Action is the slang term that guitarists use to describe the height of the strings from the fret board. This has a large effect on the playability of the instrument and to a certain degree the sound of the guitar. For example if I had a high action combined with a large amount of neck relief (for more info on that and truss rods go here) then the guitar would have a great deal of sustain, bending would be easier and the notes would ring very clearly, however I would not be able to play as fast and would need to grip tighter than usual this sort of action is usually preferred by Blues men such an Stevie Ray Vaughn, a low action on the other hand has the opposite effect to that described above and is usually favoured by shredders like Steve Vai or Satch.
Most guitarist's look for something at the lower end of the scale to make things easier on their fingers whilst keeping the sustain. There isn't really any way to see what any given action is like other than by playing a guitar set up that way. I've tried extremely low and relatively high but find that the best for me is somewhere in-between for the music that I play (rock, metal and blues).
How To Adjust It
This varies form guitar to guitar but there are 3 major categories which I will look at in turn. However with the exceptions of extreme cases action is always done at the bridge not the truss rod, I REPEAT NOT THE BLOODY TRUSS ROD!!! That is just a fast track to damaging your guitar if you don't know what you're doing; no we've got that clear here are the main styles of bridges.
01. Strat Style: this is typically 6 individual saddles with two Allen bolts in each one. The way you adjust the action is to simply turn the Allen bolts and adjust the height of the string. These allow you to adjust the action for each string individually but do take longer. This style also incorporates teles and other guitars.
02. Tune-O-Matic Style AKA Gibson Style: much quicker to adjust this is simply two thumb wheels underneath the bridge that you turn and the job-lot goes up and down, although you cannot adjust each string individually but as the bridge (should) already follow the neck radius it doesn't matter.
03. Floyd Rose: this is just about the only thing on a Floyd that is easy and doesn't take hours of practise in my opinion. Look at the tremolo and there should be two posts that the whole lot pivots on (shown in the pic) just turn these as they should be Allen bolts or screws.
This is slightly harder; it is still done at the bridge but is now done by removing the strings and taking the bridge bone out. This is the strip of bone (more often plastic nowadays though) that the strings rest on. It may not be glued in, but if it is get a hair dryer and heat the bridge up for a couple of minutes (or as long as it takes to get it loose), then carefully take it out. What you do then is to file/sand the bottom of it. The best way to sand it is to put the paper around a block and move the bridge. Remember that a little goes a long way. After you're satisfied put it back in, restring, and see what you think. Please bear in mind that this is irreversible; however if you do go too far don't worry replacements are very cheap and easy to get hold of.
Extreme Cases And Other Tips
Earlier on I mentioned that action adjustment was always done at the bridge except in extreme cases well these are they. The first is your neck having too much or too little relief (curvature) in it. I won't go into this here as it's a separate issue that I covered in my other article here.
The second is that the action is too high at the nut. If this is the case then no amount of adjustment at the bridge will make the slightest bit of difference if you are certain you have this problem (only act if you are certain) you need to either file the string slots deeper OR file the bottom of the nut. Neither is something I recommend you do. If you want to file the slots deeper buy a set of nut files and file them making sure to file at an angle with the low end on the headstock side and the high on the neck side. Do not use strings! Spend some money and do the job properly; if you use strings to file it you will probably make the holes the wrong size or make them go wonky which is not what you need.
The other way is the same as the acoustic bridge method. Just get the nut out by heating the glue. Please don't use a hammer and chisel. Brute force, bloody ignorance and guitars don't make good bed fellows as a general rulehen sand the bottom of the nut. If you've got a locking nut (Floyd style) you'll need to sand the neck. This is, however, reversible as you can make shims out of coke cans according to Powerfreak.
Oh one final point if it IS too height at the nut then the first few frets will always play out of tune no matter how much you adjust the intonation (for info on that go here)
That's your lot!