Jesus's presentation of the bread and wine as his body and blood is heavily tied up in symbolism relating to the account of the Passover in Exodus 12:

Then they shall take some of the blood and put it on the two doorposts and the lintel of the houses in which they eat it. [...] For I will pass through the land of Egypt that night, and I will strike all the firstborn in the land of Egypt, both man and beast; and on all the gods of Egypt I will execute judgments: I am the Lord. The blood shall be a sign for you, on the houses where you are. And when I see the blood, I will pass over you, and no plague will befall you to destroy you, when I strike the land of Egypt. (Exodus 12:7,12-13 ESV)

In the Exodus account, having the blood over the lintel saved the Israelites from the penalty enacted upon the Egyptians for Pharaoh's refusal to let the Israelites leave Egypt; they did not have to pay the blood price because the lamb had paid it for them. (Remember that this is at a point in Israel's history where the sacrificial system is in place; the price of sin is death, but the death of an animal may be substituted for the death of the sinner.)

Jesus is drawing on this account when he institutes what we call the Lord's Supper (remember, he is doing this at Passover so our minds should already be looking for connections):

Now as they were eating, Jesus took bread, and after blessing it broke it and gave it to the disciples, and said, “Take, eat; this is my body.” And he took a cup, and when he had given thanks he gave it to them, saying, “Drink of it, all of you, for this is my blood of the covenant, which is poured out for many for the forgiveness of sins. (Matthew 26:26-28 ESV)

See also the parallel account in Luke:

And when the hour came, he reclined at table, and the apostles with him. And he said to them, “I have earnestly desired to eat this Passover with you before I suffer. For I tell you I will not eat it until it is fulfilled in the kingdom of God.” And he took a cup, and when he had given thanks he said, “Take this, and divide it among yourselves. For I tell you that from now on I will not drink of the fruit of the vine until the kingdom of God comes.” And he took bread, and when he had given thanks, he broke it and gave it to them, saying, “This is my body, which is given for you. Do this in remembrance of me.” And likewise the cup after they had eaten, saying, “This cup that is poured out for you is the new covenant in my blood. (Luke 22:14-20 ESV)

In using the language of the "covenant", Jesus is explicitly linking his blood with the lamb's blood we have already seen in the Mosaic covenant. Christian's believe that Jesus's death -- his substitutionary atonement if you want the theological term for it -- on the world's behalf acted like the Passover lamb: he bore the penalty for sin (death) on his followers' behalf so that they might be reckoned righteous before God. I can get more into the specific theology of this if people have follow-up questions.

The atoning blood of Christ is a strong theme in other New Testament books. The author of Hebrews writes,

For since the law has but a shadow of the good things to come instead of the true form of these realities, it can never, by the same sacrifices that are continually offered every year, make perfect those who draw near. Otherwise, would they not have ceased to be offered, since the worshipers, having once been cleansed, would no longer have any consciousness of sins? But in these sacrifices there is a reminder of sins every year. For it is impossible for the blood of bulls and goats to take away sins.

[...] And by that will we have been sanctified through the offering of the body of Jesus Christ once for all. And every priest stands daily at his service, offering repeatedly the same sacrifices, which can never take away sins. But when Christ had offered for all time a single sacrifice for sins, he sat down at the right hand of God, waiting from that time until his enemies should be made a footstool for his feet. For by a single offering he has perfected for all time those who are being sanctified. (Hebrews 10:1-4, 10-14 ESV)

This is, in effect, why the early Christians (and the earliest Christians were all Jews) did not continue the sacrificial/Temple system: Christ's death and resurrection had accomplished all that the Temple system could (and more), once and for all. (It is important to remember that when Christians take Communion there is no "re-sacrifice" of Christ as is sometimes alleged particularly about denominations that believe in transubstantiation.) This language is also picked up in the apocalyptic writings of the book of Revelation:

After this I looked, and behold, a great multitude that no one could number, from every nation, from all tribes and peoples and languages, standing before the throne and before the Lamb, clothed in white robes, with palm branches in their hands, and crying out with a loud voice, “Salvation belongs to our God who sits on the throne, and to the Lamb!” And all the angels were standing around the throne and around the elders and the four living creatures, and they fell on their faces before the throne and worshiped God, saying, “Amen! Blessing and glory and wisdom and thanksgiving and honor and power and might be to our God forever and ever! Amen.”

Then one of the elders addressed me, saying, “Who are these, clothed in white robes, and from where have they come?” I said to him, “Sir, you know.” And he said to me, “These are the ones coming out of the great tribulation. They have washed their robes and made them white in the blood of the Lamb. (Revelation 7:9-14 ESV)

I think that about covers the wine/blood. As to bread/flesh, we see an earlier reference to that in the Gospel of John:

On the next day the crowd that remained on the other side of the sea saw that there had been only one boat there, and that Jesus had not entered the boat with his disciples, but that his disciples had gone away alone. Other boats from Tiberias came near the place where they had eaten the bread after the Lord had given thanks. So when the crowd saw that Jesus was not there, nor his disciples, they themselves got into the boats and went to Capernaum, seeking Jesus.

When they found him on the other side of the sea, they said to him, “Rabbi, when did you come here?” Jesus answered them, “Truly, truly, I say to you, you are seeking me, not because you saw signs, but because you ate your fill of the loaves. Do not work for the food that perishes, but for the food that endures to eternal life, which the Son of Man will give to you. For on him God the Father has set his seal.” Then they said to him, “What must we do, to be doing the works of God?” Jesus answered them, “This is the work of God, that you believe in him whom he has sent.” So they said to him, “Then what sign do you do, that we may see and believe you? What work do you perform? Our fathers ate the manna in the wilderness; as it is written, ‘He gave them bread from heaven to eat.’” Jesus then said to them, “Truly, truly, I say to you, it was not Moses who gave you the bread from heaven, but my Father gives you the true bread from heaven. For the bread of God is he who comes down from heaven and gives life to the world.” They said to him, “Sir, give us this bread always.”

Jesus said to them, “I am the bread of life; whoever comes to me shall not hunger, and whoever believes in me shall never thirst. But I said to you that you have seen me and yet do not believe. All that the Father gives me will come to me, and whoever comes to me I will never cast out. For I have come down from heaven, not to do my own will but the will of him who sent me. And this is the will of him who sent me, that I should lose nothing of all that he has given me, but raise it up on the last day. For this is the will of my Father, that everyone who looks on the Son and believes in him should have eternal life, and I will raise him up on the last day.”

So the Jews grumbled about him, because he said, “I am the bread that came down from heaven.” They said, “Is not this Jesus, the son of Joseph, whose father and mother we know? How does he now say, ‘I have come down from heaven’?” Jesus answered them, “Do not grumble among yourselves. No one can come to me unless the Father who sent me draws him. And I will raise him up on the last day. It is written in the Prophets, ‘And they will all be taught by God.’ Everyone who has heard and learned from the Father comes to me— not that anyone has seen the Father except he who is from God; he has seen the Father. Truly, truly, I say to you, whoever believes has eternal life. I am the bread of life. Your fathers ate the manna in the wilderness, and they died. This is the bread that comes down from heaven, so that one may eat of it and not die. I am the living bread that came down from heaven. If anyone eats of this bread, he will live forever. And the bread that I will give for the life of the world is my flesh.”

The Jews then disputed among themselves, saying, “How can this man give us his flesh to eat?” So Jesus said to them, “Truly, truly, I say to you, unless you eat the flesh of the Son of Man and drink his blood, you have no life in you. Whoever feeds on my flesh and drinks my blood has eternal life, and I will raise him up on the last day. For my flesh is true food, and my blood is true drink. Whoever feeds on my flesh and drinks my blood abides in me, and I in him. As the living Father sent me, and I live because of the Father, so whoever feeds on me, he also will live because of me. This is the bread that came down from heaven, not like the bread the fathers ate, and died. Whoever feeds on this bread will live forever.” Jesus said these things in the synagogue, as he taught at Capernaum. (John 6:22-59 ESV)

CONT'D BELOW:

Our knowledge of how the Lord's Supper / Eucharist was understood, theologically speaking, by the Christians in the first century is virtually limited to what we can infer from a few Biblical texts: 1 Corinthians 11:23-26 (or, more broadly, 11:17-34); Mark 14:22-25 (and its close parallels in the gospels of Matthew & Luke); and John 6 (particularly verses 48-58).

The order in which I've listed these passages is typically thought to be the chronological order in which these texts were composed; and it also seems to roughly correspond with an increasing level of detail or development of this tradition. Paul in 1 Corinthians simply mentions a tradition that was instituted by Jesus "on the night when he was betrayed" -- one in which Jesus, while celebrating an unspecified meal with here unnamed parties, identifies the Passover bread and wine with his body and blood.

The passage in the gospel of Mark, while very similar to much of the language related by Paul, is placed in a broader narrative context that recounts the wider Passover festival at the time, and the events leading up to Jesus' death; and it adds some other detail here, too -- and the same goes for Matthew and Luke, as well (both of which are nearly universally held to be literarily dependent on the gospel of Mark).

The most important element that's found in the gospel of Mark but that isn't mentioned by Paul is probably just its setting during Passover, with the disciples. However, the aspect of this meal and the act of Jesus being sacrificial -- representing the death of Jesus as what we call a substitutionary sacrifice -- is more clear in Mark, too: Jesus explicitly says that the wine is/represents his blood which is "poured out for many" (14:24). But when it comes to 1 Corinthians, this sacrificial aspect can only be inferred from the comment that the bread is/represents Jesus' body ὑπὲρ ὑμῶν, "on behalf of you" (11:24), with some of the greater ambiguities attendant on this.

(For more on the sacrificial connotation of this, though, see Fitzmyer, First Corinthians, 439. There's also a potential connection with 1 Corinthians 5:7-8, though some of this is uncertain. Finally, later manuscripts of 1 Corinthians 11:24 add a word to this line so that Jesus now says that his body is "broken for you.")

One other different element that might be significant here is that in 1 Corinthians, Jesus suggests that the Passover cup is the "new covenant," whereas in Mark his blood is merely the covenant; no "new." However, "new covenant" does reappear in the parallel in Luke 22:20. For the most part though -- again, minus the fact that the gospels of Mark, Matthew and Luke supply the broader narrative context for this Passover meal, in the course of the wider Passover festival and of Jesus' final days -- all together, we have to say that the traditions of 1 Corinthians and these three gospels are very similar.

I'll hopefully get back to how the Lord's Supper / Eucharist is portrayed in the gospel of John shortly. For now, though, the main question is how exactly was this tradition of the Lord's Supper interpreted and understood? What are its backgrounds?

Just in terms of generalities, we can say that at the very minimum that these traditions were modeled on elements of the Jewish celebration of Passover/Pesach. (Besides its explicit setting during the Passover meal as is specified in the gospels, there are any number of other details supporting this, too. Brant Pitre's Jesus and the Jewish Roots of the Eucharist is exhaustive study of this. There are also other related traditions potentially associated with it, from things like Exodus 24:8 to traditions that actually syncretized Passover with traditions pertaining to Yom Kippur.)

But it's probably safe to say that even by the time of the earliest New Testament documents which describe this ritual, it was understood in some sense as a replacement of Jewish practices, or as an expression of what's called a type of supersessionism -- the superiority of new Christian beliefs and rites over traditional ones, often associated with the non-applicability of previous Jewish rituals; or perhaps even a deliberate antagonism toward Jewish precursors.

This aspect is regularly expressed by Biblical scholars and theologians, who often point out that there's something about the fundamental spirit of the Christian Eucharist, with its clear anthropophagous and haematophagous elements, that's repulsive to traditional Jewish ideals: for a forceful statement of this, see Michael Cahill's "Drinking Blood at a Kosher Eucharist? The Sound of Scholarly Silence" -- or, a bit earlier, John Fenton's "Eating People." The latter quotes from C. G. Montefiore's 1909 The Synoptic Gospels,

I would also venture to suggest how difficult it is for us to believe that a Palestinian or Galilean Jew could have suggested that in drinking wine his disciples were, even symbolically, drinking blood.

At the same time, writing in 1991, Fenton also suggests that

None of the commentaries on the Synoptic Gospels, or on 1 Corinthians, ever seemed to address this problem; they never explained how a Jew could have said to other Jews that there was an analogy between eating bread and eating a human body, or between drinking wine and drinking human blood; and that his disciples must do both.

This bewilderment over the Jewishness or even nominal acceptability of what's said in the Eucharistic traditions has led to skepticism, and the "(not) even symbolically" qualifier has been echoed throughout scholarship over the past few decades to this effect. Thomas O'Loughlin reiterates Cahill's arguments, referred to above, that "no Jewish group, attempting to keep the laws of kosher as we know the early Jewish followers did attempt to do, could ever have contemplated drinking any cup that claimed, even symbolically, to be a cup of blood" (The Eucharist: Origins and Contemporary Understandings, 186).

Perhaps most pointedly, Maurice Casey writes, in reference to the Eucharist account in the gospel of John, that "The requirement that [Jesus' followers] should drink blood, even symbolically, shows that the eucharist has been rewritten to be as offensive as possible to 'the Jews'" (Is John's Gospel True?, 44; see also, in shorter form, his article "No Cannibals at Passover"). Roch Kereszty, with reference to Leviticus 17:11, suggests that the tradition only becomes unoffensive when Jesus' identity is understood as none other than that of God himself, because otherwise "If a mere human being commanded people to drink his own blood, even symbolically, he would commit the most hideous crime of abusing God's exclusive possession of all lives" ("The Eucharist of the Church and the One Self-Offering of Christ," 244).

On this note, the Eucharistic account in the gospel of John has several unique and interesting elements. First and foremost, unlike the gospels of Mark, Matthew & Luke, this account/discourse doesn't occur during Jesus' final hours in Jerusalem during Passover at all, but in Capernaum, long before his death. Further, whatever circumlocutions that might be used in 1 Corinthians and the other gospels to avoid some of the offense of a more direct association of flesh-eating and blood-drinking are dropped here:

So Jesus said to them, "Very truly, I tell you, unless you eat the flesh of the Son of Man and drink his blood, you have no life in you. Those who eat my flesh and drink my blood have eternal life, and I will raise them up on the last day; for my flesh is true food and my blood is true drink. (John 6:53-55, NRSV)

(For an interesting article that attempts to draw a deeper connection between early Christian-Jewish polemics or "anti-Judaism," the Eucharistic account in John 6, and ancient traditions relating to cannibalism, see J. Albert Harrill, "Cannibalistic Language in the Fourth Gospel and Greco-Roman Polemics of Factionalism [John 6:52-66].")

Ironically enough though, at the same time as the gospel of John seems to evince a more vulgar understanding of the Eucharist, it also has a closer connection than anything else to at least one prior Jewish tradition -- found in the deuterocanonical Biblical book of Sirach, a.k.a. Wisdom. Here, for example, a personified Wisdom suggests that "Those who eat me will hunger yet [for more wisdom], and those who drink me will thirst yet [for more wisdom]" (Sirach 24:21). (I've written about connections between the two in great detail here; see also perhaps Proverbs 9:2, etc. Less probable is a connection with something like Ezekiel 39:17-19 -- there'd have to have been a pretty radical reinterpretation of this text.†)

In all these discussions, there's also the looming specter of the possibility of non-Jewish Greco-Roman influence on or connections with traditions that relate to the "consumption" of gods, or theophagy. Suggestions of a connection with the later Roman cult of Mithras are almost certainly misguided; and to the extent that anyone suggests potential connections here, the focus is on traditions pertaining to the god Dionysus.

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Hit the character limit; continued in comment below.