smullen,
Pay no attention to the chattering masses. If they don't have anything "high road" to say, they shouldn't say anything at all on THIS forum.
The form itself is called BATF Form 4473, and the background check is called a NICS check (National Instant Check System). The form is the only thing you have to fill out in the state of Missouri to buy a firearm, unless you want something Full-Auto (1 trigger pull= multiple rounds fired), in which case I am completely clueless about that whole process (don't have $20K sitting around to buy something like that), but I think maybe there is some kind of 6 month waiting period and a $200 tax stamp involved. The civilian legal AR15's, AK47's, MP5's, etc. that most people here have are semi-auto (1 trigger pull= 1 round fired) and do not require anything more than a Form 4473 to be filled out.
As for the questions on Form 4473, see the link provided by jackdanson.