The headpiece thing is a coronet (literally, "little crown") and probably should be something better than brass. The cornet is, indeed, brass. It looks much like a trumpet but is smaller.
If you're really interested in the brass instruments, look at the Dynasty or Yamaha websites; they both have good illustrations and descriptions of the marching brass. Or, since you're in Denver, drop in on a Blue Knights rehearsal on the weekend, view a demonstration of various size marching brass. (Don't get in the way of the drum line.)