A mute changes the sound that the horn makes. You stick the mute into the bell of the horn and then blow normally. Sometimes you need to use more air than usual. It can distort the sound depending on the mute. Most common mutes needed are Straight Mutes and Cup Mutes. Harmon Mutes are used a lot for jazz, but I've played in wind ensembles that have used harmon mutes. Heres a picture...


http://upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Trombone_Plunger.jpg/220px-Trombone_Plunger.jpg

thats a plunger mute. Also quite common.

Yeah, the cool thing is, each mute has it's own distinct sound. I can't say I've come across a lot of call for harmon mutes, even in the big bands I have played with. I do, however, find that in big band I end up with a bucket mute in my bell quite a bit.
Unfortunately, if you're gonna do a lot of bone playing, you'll need quite a few to be prepared. A straight and cup and a plunger will get you far. Eventually, you may want to add a bucket, harmon, pixie, or derby, depending on what you're playing.
One of my biggest pet peeves is someone showing up unprepared and using a cup mute where a straight mute is called for, etc. It just doesn't blend. Use the right tool for the job!!

You may also run across a chart that calls for you to use your left hand as a mute. If you see "hand" as a notation expirement with your instructor and your hand to arrive at the sound that's called for. The notation "hat" which originally meant a derby style hat used as a mute is out moded. But I have seen guys fold a fedora in half and use it to the same effect. (There was a time when no self respecting man would venture out in public without a hat.)

Funny thing is, I actually own a derby. But I have never played in a section where, when it was called for, everyone in the section had one. So, for the sake of consistency, we'd all usually just play into the stand. So it has never gotten to be used.