AP: This gospel is just a scientifically reconstructed Gospel. It’s a hypothetical document since no existing manuscript has been found thus far. Nevertheless, it’s a well founded hypothesis.
The "Q Source" refers to those passages in Matthew and Luke that match significantly (at times literally) and are not found in the Gospel of Mark, which Matthew and Mark copied as one of their sources.
These passages in Matthew and Luke, some of which are very long, look so similar that scholars think that such similarities were not the result of oral tradition. There must have been a Gospel already in existence that contained the “sayings of the Lord.” Matthew and Luke must have had this "Gospel" in front of them and they copied those parts that interested them.
But it should be noted that no manuscripts of that Gospel have been found. However, most New Testament scholars think it is highly probable that it existed and it is just mere coincidence that we have yet to find any manuscripts to confirm its existence.
TWH: Antonio gives a really good description about what the "Q" source is. Allow me to just offer a few thoughts concerning its nature and its value for New Testament studies.
This is a hypothetical document, just as Antonio said. Not one single manuscript to date has been found supporting its existence. In other words, some scholars believe it must have existed not because they have external evidence in favor of its existence, but because they are attempting to explain similarities/differences between the Synoptic Gospels. In doing so, they use internal evidence exclusively. Personally, I find the argument for Q (and all other hypothetical Gospel documents [e.g., L, M, etc.]) to be unconvincing.
"Q" is an attempt to explain the origins of the Gospels. I think the starting point is flawed. The general scholarly consensus today is that Mark wrote his Gospel first. The external evidence, however, favors Matthew's Gospel as the first. While there are a number of arguments given to prove Markan priority, let me just give you one:
Mark does not include the Sermon on the Mount, for example. It seems unlikely that Mark would cut out such a significant portion of Jesus' teaching. Rather, say those who favor Markan priority, it is more likely that Matthew wrote and expanded Mark's Gospel.The problem with such an argument is two-fold. First, the external evidence is clear: Matthew wrote first. Second, the external evidence tells us that Mark's Gospel is the collection of Peter's messages in Rome. Peter's purpose was not to give an exhaustive account of Jesus' teachings. Rather, he wanted to give people in Rome an energetic presentation of Jesus' life and ministry, paying special attention to the the wondrous works Jesus did. If that's the case, it would not be surprising for Peter to not include the Sermon on the Mount. The interest in "Q" is directly tied to the general scholarly consensus that Mark wrote first. Hopefully, in the years ahead, scholars will rethink whether or not the contents of the Gospels (similarities and dissimilarities) can be explained apart from any hypothetical documents. I think if they do, what they will end up finding is an explanation of the origins of the Gospels supported by external evidence, i.e., the writings of the early church.
If you want to see just how serious people are about the "Q" source, do a Google book or Amazon search on it. It's pretty amazing really. I mean there are whole commentaries written on this hypothetical document. I'm not sure how people know so much about something that has so little an imprint (in my opinion, zero) in the historical record.