This one's pretty easy to wrap our minds around. The title lets us know that the story we're about to read is a gospel (see "Genre") and that it was written by someone named Matthew. Ta-da!
Christian tradition has always regarded this Matthew as one of the twelve disciples (check him out in 9:9). Of course, this theory doesn't really make much sense. Why would one of the original disciples, an eyewitness to the life of Jesus, have copied nearly 90% of his stories from Mark and other sources (source)? You'd think he'd have tales of his own to tell. And that they'd be pretty good ones.
What's clear at least is that the author was a Jewish Christian. He quotes from the Hebrew Bible all over the place, knows his Torah inside and out, and has really specific thoughts about different Jewish groups. But we'll probably never know exactly who he was, how he became Christian, or what his relationship to Jesus was (source).
Looks like the "Matthew" of our title is a bit of a mystery.