Even when the guitar is unplugged, you shouldn't hear much "buzzing" unless you're REALLY strumming on it. The harder you strum, the more the strings vibrate... which means the more space they need to vibrate. There are several factors that can contribute to fret buzz and several ways to remedy it... The first step would be to identify your ideal set up, then adjust the guitar accordingly.

String guage (size) is one of the first things I would look at. If you're using really light gauge (small) strings, the strings will take less tension to reach their standard tuning... meaning the elliptical pattern of their vibrations will be more broad. Basically, heavy guage strings vibrate in a tighter pattern than lighter guage strings and offer more "buzz" free benefits.

Truss rod... The truss rod is a nut and rod system that runs through the neck. This system is used to counter the string tension on the neck. If you detune your guitar or change from light to heavier strings you might notice that the strings are higher or lower to the fretboard depending on what strings you change to. If you intsall heavier strings, there will be more tension on the neck, if you intstall lighter strings there will be less tension on the neck. Without going into to detail about adjusting your trussrod, I'll just say that if you look down the edge of the fretboard from the bridge toward the headstock... The neck should appear almost flat with a SLIGHT curve upward (toward the strings) around the 5-7 fret. This is called "relief". If your neck looks completely straight or curving downward (away from the strings) then you need to adjust this.

There are cases where no matter how much you adjust a truss rod, there still seems to be some buzz... This can be a poor fret dressing or installation. This is pretty commong on cheaper instruments since the fretting process is such an involved process, it's often hurried or done by less accurate craftsmen and can leave a lot of inconsistencies...

Bridge height... Your guitar should ahve one of two types of bridge adjustments. Either it has individual saddles that have height adjustment screws or a tailpiece that has height adjustments on the mounting posts... If your truss rod is set properly and you're still getting fret buzz, you may need to raise your bridge/saddles. Remember that if you adjust the truss rod or saddles that you're changing the distance between the bridge and the nut... This will alter your intonation... search for that in these forums and you can learn all about it. I believe I have explained how to adjust it before...

Chances are, if you're getting a lot of buzz with just an open E string... try a heavier guage string first.