From classical, jazz, to flamenco and folk; each style and tradition calls it differently, and sometimes even approach it differently. Why is it used instead of normal strokes?
Conceptually, what is it? And tonally, how does it affect the sound?
what is it?
In classical guitar technique, a rest stroke is when your finger plucks the string and comes to rest -- hence the name -- on the next string over. I assume the idea is the same regardless of which style you're playing.
how does it affect the sound?
Why is it used instead of normal strokes?
As mentioned, this is a technique for finger style classical guitar.
There are two types of stroke:
tirando, or free stroke
apoyando, or rest stroke
Both are valid techniques and guitarists are encouraged to learn both. Some people think that rest stroke is "easier" but in realty neither is easy for a beginner and either can produce a strong full volume sound with practice. Experienced guitarists can get the same volume from free stroke as with rest stroke.
The use of these has changed with history. For a while the rest stroke was favored then fell out of favor with free stroke rising. Some flamenco guitarists teach that the top note of any melodic run should be attacked with the rest stroke with the other free stroke. I'm sure people have their reasons for such rules but in my experience this is a personal choice and does not necessarily produce better sound.
One does not pick the string by pulling it parallel to the top but pushing the string in a little. The string rolls off the nail in an elliptical arc. This puts a lot of potential energy into the string leading to large amplitude movement. There is only so much you can displace a string before you create buzzing from the string slapping on the frets so over driving the string with a heavy rest stroke can be counter productive. Many modern classical guitarists are of the opinion that the rest stroke should be avoided at all costs. In Parkening's beginner book he advises students to play every exercise using both free and rest stroke. Segovia's scale exercises are introduced with rest stroke only, but most do them both ways. I have seen some flamenco tremolo exercises where they recommend doing tremolo with rest stroke just to train the fingers to feel where they should be (or not be) then tell the student to transition to free stroke.
As for volume and tone. I can say from personal experience that I can play free as loud as rest and the tone is better. Sometimes you want the next string to vibrate to support harmonics of the note being played. In this regard rest stroke can cause a slightly dead sound. As for volume, I can definitely put more force into the string with rest but only at the level where it overdrives the string and causes buzzing.
Conceptually, it's playing a particular string, and stopping the stroke as the plucking finger reaches the next string - generally the one higher, physically, on the guitar. Thumb obviously works the opposite way.
Tonally, I don't think it makes much if any difference.
It's useful in that the finger doesn't stray far from the strings, and if the next note is on that next string, it's ideally placed to play it. I don't think the damping effect is particularly important - the other string maybe shouldn't have been sounding anyway, and there's always the fretting hand to mute it.
For beginner solo players, it does help to keep the hand engaged with the strings. I find a lot of beginners 'float' about with their hand, with and without pick, so there's a tendency to pluck a wrong string. Rest strokes will help avoid that.
I catch myself using rest strokes on bass guitar in some numbers - and I can't figure out why! It's not by design, just something that's developed. When the penny drops, I'll let you know!