What is action/ Why do I care
Action is fancy guitar talk for how high the strings are suspended above the fret board. If you have too high of action the guitar can be very stiff to play. Too low and you'll get buzzing strings. Apart from the obviously too low and obviously too high where your action lies is all about personal preference though in general a lower action is much easier to play.
A very low action can make it easier to do that fast fancy fretwork stuff. For example your jazz and rock guitarists like their action on the low end. Jazz can compensate the potential for buzz by having a heavier gauge string whereas rock uses incredibly light gauge (got to do those bendy things) but are able to compensate for some buzz with distortion and guitar effects.
You might want a more average action if you are doing rhythm guitar, if you are playing a more clean sound (as opposed to distorted) or for practice as strings that are harder to press are probably better for building up that finger strength (though you do you). You'll want high action (in fact it is strictly necessary) if you play with a slide.
before you touch anything
Keep in mind that action is effected by your truss rod and the bowing of the guitar neck. If your guitar neck doesn't have the proper bow then your action will be all over the place and you don't want to go adjusting it till you have that fixed. I talked in my other post about some of the problems with a bowing neck so check that out.
How to adjust action (for electrics)
So its all depending on the type of bridge. Take a look here for a visual guide of the some bridges.
Gibson/Floyd Rose like bridges
In general if you have the Gibson style ones "tune-o-matic" or the "wrap around" you will see two large thumb wheels on both sides. This if for adjusting the action. As you can imagine tightening the wheels lowers the action lossening highers the action.
The floyd rose is similar though you need an allen wrench. But it is also adjusted using the two larger screws on each side.
If you want to do this without a feeler gauge but as close to the official way as possible you can use four common objects as your feeler gauge. A Heavy guitar pick, A thin guitar pick, a dime and a nickel. Esentially you want your high E to have a distance equal to a thin pick between the first fret (the actual metal bit) and the string. So stick the thin pick there and take note of how it fits and then repeat with your dime at the twelfth fret. and for low E do the heavy pick and then the Nickel. Then adjust accordingly if your objects had lots of space bring the action down. If it didn't fit you can consider bringing it up though these objects get you a good average action so going lower isn't necessarily a problem.
Fenders (and guitars based on Fenders)
So on the fender style bridges "hard tail" "fender tremolo" and slightly different but the "telecaster" each string has action set independently. To adjust it you need to look at the saddle (this is the metal place that the string rests on just after coming out of the back of the guitar). On each and every saddle are two tiny little screws. These take a tiny allen wrench. Now its vital that both screws on a single saddle are at the same height. So if one is lowered the other needs to be lowered the same amount.
Doing it the right way is more complicated because each and every string has its action set independently. I don't have a clever hack for you. I just eyeball everything because anything worth doing is worth doing wrong. The right way to do it is to make those same measurements you would on the gibson (with the picks and coins) to get your low E and high E set up and then you have to have whats called a radius gauge (its a special luthier tool) and you set that up to rest on your e strings to see if the strings have the right curvature. and then adjust from there
What I do is just adjust it till it feels good and then play it to make sure it doesn't sound like shit.
Anyway thats a bit about action. I might talk about adjusting it on Acoustics (which is a whole different beast) and I still got to talk about intonation. So maybe in a couple days if I feel up to it.