| What they are and when to use them are really two different things.
The 6th chord is just a major or minor chord with a note that is a 6th above the root added in. You'll likely hear them more on major chords than minor. On minor they tend to make a diminished sound, on major, an added 'sweetness'. The last chord of The Beatles "She Loves You" has an added 6th. A bit unusual for pop music in that time, most it was heard on jazz tunes.
As for the other chord extentions, 9th, 11th, 13th. Play them on the piano (guitar voicings often omit certain notes in extended harmony) and get the sound in your head... then try them out in some music and see what you think. Generally speaking (and I'm sure there will be some comments on this) when you think of a 9th chord, there is an assumption that a 7th is also present (if you don't want that, an accepted way of notating this to put a 2 after the root letter.. C2 would be C, E G, D). An 11th chord assumes there is a 7th and a 9th, etc etc. Remember too, that a 9th, 11th and 13th can be altered away from the key signature, just like the 7th. You can have major or minor 7ths, major or minor 9th (very cool sound), 11ths are the same as a 4th (a perfect interval from the root) so technically should be diminished, perfect or augmented, and the 13th is the same as the 6th, so... minor, major.
All of the technical and theoretical words and knowledge are good for helping remember how things are built and what they are... but it is your ear that should be the altimate judge.
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Never confuse beauty with things that put your mind at ease. -Charles E. Ives
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