How do I get rid of fret buzz?

Look to the strings first. You may need to change them.

Are they accumulating a film? (You may need to look on the underside to see all the dirt you have on them.) There are a handful of products designed to extend the life of your strings, but if you want to go the economy route, simply wiping them down with a slightly damp cloth will be better than nothing.

Do you see dirt and grime building up on the fret board? (Minor, but if you're cleaning for grime, you might as well clean this surface during strings changes.) You may want to clean this using a slightly damp cloth or a good guitar cleaner. Some people recommend using the finest grade of steel wool to abrade extra tough dirt off your fret board. I haven't had to go this far yet, and would urge caution to go easy on the elbow grease as to not scratch the finish on the wood.

Long sweat and dirt buildup can eventually damage glue joints and warp your instrument (this would be after a long period of exposure without any maintenance). I'm by no means an avid "guitar-polishing guy" but the occasional cleaning of the fret board will extend its life.

If this doesn't fix your problem, the solution may require the expertise of an experienced technician. There are some things you should do to pinpoint the location of your buzz, but I'd never recommend someone grab their tools and try to adjust a guitar on their own. Learn from someone with experience and your guitar won't become a field casualty.

Check to see if the buzz is consistent across the length of the neck (from open strings to the frets with small spacing towards the body of the instrument).

Pick each string without fretting any notes.

Fret each string at the first fret and move towards the body. You may not need to successively fret each position, but if you're compulsive about your instrument, then you can check each fret.

Make a note as to where the buzz appears and where is goes away. If the buzz stays from the open strings and doesn't go away even when you're fretting the highest pitched frets, the problem is likely in your bridge/saddle assembly. Are there any loose parts? You may need to have a technician work on this area. See if they'll let you view the process. I've gleaned a lot of maintenance knowledge by being present and asking simple question while someone worked on my instruments. (Don't get in the way, technicians will tolerate your presence and questions, if they aren't annoyed by your presence.)

If you pinpoint the buzzing to the section of your neck closer to the body:

You may need to have your strings saddles adjusted up so they will be farther away from the first frets they come across. If you do this yourself, try to make your adjustments in slight increments so you don't raise your action more than you need to. If the height of the saddle won't compensate enough, a truss rod adjustment is probably needed (see below).

There may be a fret that needs to be sanded slightly to eliminate an inconsistent span. Don't make this adjustment yourself. If you go too far, the mutilated fret will become permanently sharp in pitch.

If the buzzing is closer to the middle of the neck or towards the nut:

Inserting a slight shim under the nut can raise the strings high enough to eliminate the strings' unwelcome contact with the frets. This might be difficult if you have a glued in nut, but those of you who put up with the locking variety can easily take the nut off and insert a small piece of paper or foil under the nut. Remember to take it in increments; it's hard to play those neoclassical licks with high action.

Your neck may need to have the truss rod adjusted. The truss rod is a rod of metal that runs the length of your guitar's neck. By turning it to the left or right, you cause the wood to bow in or out. Of all the adjustments mentioned, this is the most dangerous and should definitely be handled by a technician. If you turn the rod too far (even a fraction of a turn), you can splinter the neck. Here is a step to watch and learn. Minute adjustments can make a big difference in the bow of the neck.

Some adjustments may necessitate other adjustments to preserve the action. Cleaning the neck and getting a new set of strings might solve your problems. If you're a strummer and using light strings, maybe you should raise the action and/or move up to a heavier gauge of string. Don't be afraid to take your instrument to a technician. You can learn quite a bit for future maintenance.