You can damp strings in a variety of ways. Most often the guitarist will release pressure on the strings that are being held by fingers producing the chords. The pressure is released just enough to return the strings to their normal height, but your fingers do not completely release the strings. This method works well with chords that do not use open strings. You can damp additional strings that are open by not only releasing the fretted strings, but by using those fingers to touch additional strings. In some cases you can release the fretted strings and then slightly angle your fingers so they touch adjacent strings slightly to quiet them.
Another method to dampen open strings is to use part of your right (pick) hand to dampen the strings. This is very common in many forms of Spanish and Flamingo music. Sometimes strings that have not been strummed or plucked still start to vibrate because of “sympathetic” vibrations of the strings that have been played. Sometimes this sympathetic vibration will be heard at an octave higher than you might expect. In some cases your left hand thumb can be brought around the neck just enough to dampen the sixth or even the fifth strings if needed.
In some cases you may want to dampen strings even as you pick single notes, this is usually done with the heel of the right hand while you also pick the string. You might also practice playing a “flat pick” song normally and then completely but lightly dampened. It makes a great new sound for the same song. I have done this for many country type songs.
A significantly more difficult technique is very similar to damping which is creating “bell tones”. Most guitarists have some experience with doing this while they tune a guitar, because bell tones can help you to get a guitar into a more accurate tuned condition if you don’t have an electronic tuner. Typically bell tones are produced by lightly touching the string at the 5th, 7th, or 12th fret just after picking the string. In this case you are not damping, but you are causing only the upper harmonic of the string to sound. The technique can be expanded to every location on the guitar, but you must pick the string with your right hand while holding the pick with an outstretched index finger. It is the index finger that will be used to slightly touch the string at specific points along the neck of the guitar. Thes points are proportional to the length of the string.
If for example you are holding the first string at the third fret (G note), then there are harmonic bell tones that you could play at the point where the string would be one half the way toward the bridge, but now this is three frets higher than the G note you are holding. This proportion holds for the 5th, and 7th locations also meaning they would be 5 frets higher than the G note you are holding and 7 frets higher than the G note you are holding. The 5th and 7th locations produce weaker bell tones that the 12th fret version.
If you are holding a typical C chord starting at the first fret of the second string, you could in fact strum the guitar in a pattern similar to the shape of the C chord three frets higher than the 12th fret. This is very tricky and you do dampen some strings that you do not want to sound in the process, but it gives you many more bell tones to potentially play.
Damping is particularly important in jazz which often wants to stop one particular sound before starting another sound. Damping has a lot to offer the guitarist that can master the technique.