Fret buzz is one of the most frustrating problems you can have when you first learn to play the guitar. Part of the problem is that there are so many possible causes, some of which are easily fixed, and others, less so. It makes sense then, to first of all try and identify what’s causing the problem and eliminate easily fixed causes first.
The first thing to check, if your guitar is electric, like, say a Gibson or Epiphone Les Paul, is that it’s the guitar itself that is buzzing and not the electrics. Unplug your guitar from the amp and check if the buzz is still there. Now check that the strings are seated properly in the nut (at the top of the neck) and the bridge. On old guitars, the nut, especially if its plastic, can get worn down and cause fret buzz.
If none of that is causing the problem, hold the guitar up to eye level and look across the strings, level with the pick-ups. Are the pick-ups touching the strings? If so, that could also cause buzzing.
Next, check your strings. If they’re grubby or have a film of sweat on them, that could be causing buzzing. Clean them with a damp cloth, or use a specialist cleaning product like GHC Fast Fret. The fretboard itself can also get grubby. Sweat and dirt builds up over time on the fretboard and next to the frets themselves. Clean that too, preferably with a cleaner designed for the job. If not, a damp cloth is better than nothing.
If none of that works, you may have to get professional help from a guitar tech. The cause could be a fret that’s not level with the others and needs sanding, or a saddle that needs adjustment. It could also be a worn nut that needs to be heightened or replaced, or a truss rod that needs to be adjusted.
Clean the fretboard, replace the strings and make all the checks outlined above. If none of those work, take your guitar to your local guitar store and ask their technician to take a look. It shouldn’t cost too much to make the necessary adjustments. While you’re there you might want to ask him to do a complete set-up for you. That way you’ll know that your guitar is in the best condition it can possibly be.