This is a question which is still very hotly debated. I'm going to give the more generally accepted answers, but be aware that there are large, vocal minorities of scholars who don't agree. We have no extant manuscript copies of any of the gospels before the late 2nd century, IIRC.

Mark is considered the oldest of the canonical gospels. It is usually dated to before 70 AD, as it contains no references to the destruction of the Temple in that year. Arguments from absence are, however, tricky. Matthew and Luke are usually dated in the next decade or so, between 70 and 80. Matthew shows many signs which make scholars believe that the gospel's author knows of the destruction of the Temple. Luke contains similar indications, but the eschatological fervor in the book is somewhat less than we find in Matthew, leading many to believe it was composed later, but this isn't particularly firm either.

The relative dating of the three synoptic gospels is also argued by the various intertexts we find between them. Arguments along these lines are known as the "Two Source" or "Three Source" hypothesis. Essentially, both these theories utilize the fact that Matthew and Luke are both aware of an unrecorded collection known as Q (German: Quellen - source) and Mark. It is, however, unclear from this if the author of Luke is or is not aware of the existence of Matthew, thus the "Two" or "Three" sources.

The dating for John is the most contentious of the canonical gospels. Traditional dating places it ca. 90 AD, but some (including one of my professors) want to push it as far as 150 AD.

I will leave aside the dating question, /u/telkanuru as usual has done a good job. Instead, I want to address the relationship between dating manuscripts and dating their contents.

There are different techniques for dating physical documents and for dating their literary contents. For example, the earliest copy of Plato we have is from 900 AD.

Most dating of manuscripts is done via paleography, studying the style of writing, and matching it against known quantities. This is, for the most part, more accurate than radio-carbon dating which gives broader ranges and requires some destruction of the physical source.

How do you date the contents of a manuscript though?

1. You have to date a text after anything it mentions, because naturally people can't write about things that haven't happened.
2. A text can generally be dated before any other text that clearly mentions or deliberately quotes from this text.
3. You can do some dating based on linguistic features and analysis (essentially the linguistic and literary equivalent to paleography, noting changes in language features, vocabulary, style, etc..).
4. If you assume that it is not a forgery, you can date it with reference to its historical contents and contexts.

1 & 2 actually get you a long way, since there is a decent amount of early Christian writings from the 2nd century that put the NT documents into the first. You can also cross-reference 3 with dating of manuscripts, since not all of our manuscripts are 'literary texts'. Plenty of papyri are letters, accounts, etc., which are more likely one-off writings that can be dated with paleography, but then their contents used for linguistic type dating.

So the recent renewed kerfuffle over the "Jesus' wife fragment". Radio-carbon, chemical and paleographical dating put the material of the document into the Islamic period, 7th-10th century. It could, despite this, be a forgery, because simply using old papyrus and indeterminate ink doesn't make it genuine.

However, the contents could be dated differently. This is what Professor King is arguing, that the text reflects an early, Christian, writing from the 3rd or 4th century.

Anyway, my point is that there are ways to date things before the age of the manuscript, indeed you must, otherwise you are left with Plato writing in 900, Aristotle in 1100, and about the only ancient Greek writing you can be assured of would be Homer in 400 BC, which would still be 500 years too late.