Quincey Morris is the character that we know least about... besides the fact that he's as 'Murican as apple pie.
He's first introduced to us when Lucy describes his proposal to her in a letter to Mina. Lucy says that he "is really well educated and has exquisite manners" (5.7), even though he can be super funny. He only uses American slang when no uptight English people are around to be shocked and horrified by his language. (The modern American reader will think Quincey's "American slang" is hilarious.) When Lucy rejects his proposal, he tells her that her "honesty and pluck have made [him] a friend, and that's rarer than a lover; it's more unselfish anyhow" (5.13).
Most of what we know about Quincey, besides his use of "American slang," is that he values "honesty and pluck" above almost anything. He's straightforward and never says anything he doesn't mean. He takes what people say to him at face value and accepts even the strangest things (like the existence of vampires) with remarkable calm.
In addition, Quincey is always the first person to volunteer when something needs to be shot or killed. During a meeting with the Crew of Light, he excuses himself from the table to go take a pot shot at a bat he sees from the window (18.44). Of course, we're supposed to assume it was Dracula, in the form of a bat, eavesdropping on them. None of the other characters would have taken that kind of initiative.
If you want to learn more about Quincey's death and the part he plays in destroying Dracula, check out "What's Up with the Ending?"