AP: This Gospel is only a scientific reconstruction (a hypothetical text), though very well founded. Why hypothetical? Well, simply because no extant manuscript of this source has been found so far. The "Q" source refers to the overlapping passages of Matthew and Luke that are not in the Gospel of Mark, of which is one of the written sources Matthew and Luke made use.
Where Matthew and Luke agree in those passages, which are sometimes quite lengthy, scholars think there was something more going on than just an existing oral tradition. Building on that, they suggest that there was already in existence a "Gospel" that contained primarily the "sayings of the Lord" written in the Greek language. This "Gospel" had to be the source that Matthew and Luke had their hands on, and the one that they made use of in addition to the Gospel of Mark.
But don't forget: Not a single manuscript of this Gospel has ever been found. However, the majority of New Testament scholars think that it is highly probable that it existed. That there is no extant copy today is simply a matter of coincidence.
TWH: I'll just throw out a couple of thoughts regarding this whole issue. I think it is quite significant that there is no extant copy. That's a major blow to the whole issue of whether or not the source ever existed. Still, that's not the only factor that leads me away from Q. There are other historical scenarios that are much, much, much more likely. In fact, if we just take the external evidence that we do have, we can account for the origins of the Gospels–Matthew, Mark, Luke, and John. Part of the problem stems from the modern scholarly consensus in Markan priority, a position that totally devalues and generally ignores patristic testimony. The need for Q really hinges on the need to explain how Matthew and Luke have these sections that Mark doesn't have, yet Matthew and Luke have them in common. If we understand Luke as making use of Matthew, and Peter (i.e., the Gospel of Mark) making use of Matthew and Luke when he preached in Rome, the issue really disappears.