I am in the market for an air nailer to work on a few projects. What is
the difference between a finish nailer and a brad nailer? Why would I
choose either? Sorry if this is an easy question, but I just want some
input before I go buy one.

There's been a few recent threads on this, and you might want to search back
a bit and read the threads for more detail. Basically a brad nailer shoots
18ga. nails up to 2" long. Thin nails with minimal holding power, they're
good for light moulding or holding things together until the glue dries. One
advantage is a relatively small hole

15ga and 16ga nailers are considered finish nailers. They both shoot nails
up to 2.5", and that extra length comes in handy for hanging crown, jambs,
casings, and heavier moulding. The 16ga nailers have a horizontal magazine
and the 15ga nailers have an angled magazine (with a few exceptions) making
it easier to reach into tight spots. The 15ga nail, being a bit sturdier
than the 16ga makes it more of an all-around gun. A good combination would
be an 18ga 2" brad nailer and a 15ga finish nailer.

What are some good moderly priced nailers?

You can choose from cheap no name Asian imports as sold by Harbor Freight
and others. Sometimes they work well, sometimes not. You're usually better
off with a mid-range brand such as Porter Cable, DeWalt, Bostich, Hitachi,
etc..