We've learned such a huge amount of things, both large and small, that I'm sure I'm going to forget a number of them. But here goes:
When the Scrolls were discovered and dated (using Paleography, C-14, and Mass Spectrometry) to between the 3rd century BCE and the 1st century CE (some are older, some newer), we were finally able to see if the text we had was relatively reliable. In most cases, on the macro scale, it is. There are a ton of minor textual variants (sentences, words, etc.), but you won't see the copies of Exodus saying that Pharaoh actually killed Moses and led the people out of Israel himself, if you catch my meaning. So on that level, we learned a lot about the transmission of the Bible.
We learned a ton about the variants of Second Temple Jews. While there's still scholarly debate on who exactly wrote the Scrolls (another question for another time), the general consensus is that it was the Essenes, of whom Josephus spoke. With the discovery of the Community Rule document, we got the smallest details of the way these people operated, including their initiation rites, their day to day rules, the way they used their money, and the way they interacted with each other.
The Scrolls have been a huge discovery in terms of the development of canon. For those who say the canon was closed since its original creation, the Scrolls have both supported some of their claims and muddied up the waters a bit. In terms of supporting the canon, we see that, as far back as the 3rd-2nd century BCE, the general outline of the Hebrew Bible was similar to today. They had the "writings of Moses" (Pentateuch), "The Prophets" (the major and minor prophets of our Bible), and the "Writings" (Wisdom literature and other assorted things like Job). We also have scrolls that contain both Genesis and Exodus, others with Exodus and Leviticus, Leviticus and Numbers, Numbers and Deuteronomy, so we know that they were thought of as a unit at that time. We also know that these books, along with the Psalms, were the most important to this group: there are about 40 copies of Psalms, 35 of Genesis, and similar amounts for the others, followed by a sizeable drop off when it comes to other books. For instance, there's only one of Ezra, none of Esther, and many others in the single digits.
How did it muddy up the canon? Among the most frequently found books are the Book of Jubilees and the Book of Enoch, indicating that these, too, were considered sacred, and more sacred than the majority of non-Pentateuchal books. Their writings frequently invoke ideas from the Book of Enoch and the book of Jubilees, and their calendar seemed to run by that of Jubilees.
We discovered a number of methods of biblical interpretation that are extremely interesting, and predate Midrash and Talmud. One method is the Pesher, in which verses from the Bible are laid out and the interpreter writes, one by one, how each verse applies to their modern situation (they were heavy into eschatology, believing there was going to be a massive war between the forces of good and evil). The other was in the form of pseudepigrapha–stories written from the point of view of a famous biblical character, that retell or expand on their story in order to smooth out some issues with the text. For a good example, I'd suggest the Genesis Apocryphon.
I don't know quite how to put this next one, except to say that they operated, literarily, in a very bizarre way. Part of the reason that they weren't particularly easy to date was that they never use proper names. They only use code names, and there are a number of characters that show up. Their initial leader, whom they see as something of a prophet, is only known as the Teacher of Righteousness. There's also the Man of the Lie, the Wicked Priest (his enemy), Those Who Speak Smooth Things, and a number of others that I can't recall offhand. The Teacher of Righteousness is thought to have written a number of the nonbiblical scrolls, most notably the Thanksgiving Hymns, which are a collection of really beautiful psalm-style poems. I really suggest reading them.
For Christians who are interested in the context that Jesus came out of, the Scrolls set the stage for a lot of things that appear in the New Testament. The New Jerusalem Text and Temple Scroll will remind you of Revelation, and a number of narrative devices ("fount of living water" for one) will look familiar. That's NOT to say that the Dead Sea sect were early Christians, or that Jesus was an Essene (though some people have argued weakly), but it simply shows some context. Personally, given the style and the narrative devices, I do think that Johannine christianity may have come out of a Jewish context that had some kind of influence from this group (for more on that, you can read a number of books and conference proceedings on Qumran and John), though that's as far as I'll go on that matter.
I'm missing a million things here. We learned so many little things from the Dead Sea Scrolls, along with the big things, that it literally takes whole books to explain. The best book, easiest to understand, that I've come across is The Meaning of the Dead Sea Scrolls, by Flint and Vanderkam. That's where I got all of this information.
Hope this helps!
Edit: Minor textual changes, just like the Scrolls themselves.
Edit 2: Bonus fun fact: if you're looking for a very specific example of something that we didn't know that we know now, here's one. There's a moment in 1 Samuel 11, in which a character behaves very strangely. Nahash the Ammonite makes peace with the Israelites, and one verse later gouges out everyone's right eye for seemingly no reason. This has puzzled a lot of commentators. However, scroll 4QSam-a (4=Cave 4, Q=Qumran, Sam=Samuel, a=fragment A) has an additional verse that explains Nahash's behaviour (though he still comes off as a horrible tyrant): he liked gouging people's eyes out and the Israelites had escaped and conglomerated in the city that he oppresses in the following verse. Now, the NRSV translation, which is the one used by most academic scholars, includes that verse that was originally missing. So that's a very tangible example of things we learn from the Scrolls for you.
Edit 3: Thanks for the gold! Y'all know how to make a poor grad student feel special!
I'm on mobile, so forgive me for formatting and some brevity. I'll come back and fill this out more later.
First and foremost, the DSS are the oldest extant manuscripts of the texts of the Hebrew Bible. There are also a slew of sectarian documents that deal with (most likely) the life of the community of Jews who lived at Qumran.
So what did we learn? First, we gained a great deal more insight into the practice of Textual Criticism of the Hebrew Bible. We discovered a vast number of new textual variants (and even in some cases, like that of the book of Jeremiah, we discovered wholly distinct versions of some books). One of my favorite examples comes from one of the Deuteronomy scrolls (I can't remember the call name off the top of my head - perhaps 4QDeut^j EDIT: I was right! http://ohb.berkeley.edu/Deut%2032%20sample.pdf). In this particular example, Deut 32:8-9, an old poem about Canaanite deities, Elyon ("the Most High") is dividing up the land and doling it out to junior deities, of which Yahweh is one. In the scroll, we are told that Elyon is dividing up the people's according to the "sons of God." The Masoretic Text of the Hebrew Bible says "sons of Israel" here. The reading "sons of god" makes much better sense in context and so we can see that someone changed the poem at some point as a function of their monotheistic piety.
Additional information added to my original comment:
Basically, we learned a great deal about the development of many of the texts composing the Hebrew Bible canon. We learned more about how at least one sectarian community maintained several different traditions of some texts, we learned that they even used older forms of the Hebrew script (e.g., 4QPalaeoExod). We learned that were comfortable with a pluriformity of text types.
Double Ninja Edit, now with sources!!
So if you want to READ THE SCROLLS! check out The Dead Sea Scrolls Study Edition.
If you want a history of their discovery and scholarship, check out Lawrence Schiffman's Reclaiming the Dead Sea Scrolls.
If you want to check out some of the scrolls online! go here: http://www.deadseascrolls.org.il/explore-the-archive
Brill has a series of Dead Sea Scrolls Readers that are really great (albeit pricey).
Jodi Magness is the renown archaeologist for Kh. Qumran and her work is excellent.
We also learned a great deal (although much is still admittedly mysterious) about the khirbet around which the caves were situated. We found more in wells there than we really ever have elsewhere.
I'll more a few more updates, including a list of invaluable DSS resources in about ten minutes.
ANOTHER EDIT:
EVERYONE SHOULD KNOW THAT WE HAVE RECENTLY DISCOVERED A NEW DEAD SEA SCROLL CAVE. LIKE LITERALLY THIS IS A NEW DISCOVERY AND WE'RE ALL A LITTLE EXCITED ABOUT IT.
http://www.cnn.com/2017/02/08/world/new-dead-sea-scrolls-cave-discovered/index.html
HERE'S A PICTURE:
As a followup question, I was raised in your fairly typical Christian household, and I was always told that the Dead Sea Scrolls disproved the common idea that "There's no way The Bible is accurate, since it's so old it must have lost so much in translation over 1000's of years". I was told that Bible scholars studied the book of Isaiah for example, and found that the only differences to a modern text were very minor grammatical errors or use of synonyms, and nothing that actually changed any meaning. Is this true? Or is this a myth/falsehood that's just widely spread because a lot of people don't know better?
Forgive me for my stupid question, but is the Bible an extension of the scroll, or are they different?
As informative as the informations above are, I can't help but feeling more and more confused. With every word I read, I'm like "huh??" "What??" I'm sorry 😐
Also, book of Samuel talks about the rise of David, but have you noticed that there is a possibility that there are more than 1 David? E.g., there is David the shepherd, David the brat (judging from the way he talks to his brothers), David the warrior, David the lye (sp?) player who is able to cast off evil spirit off Saul.
