The 6th chord is a major chord with an additional major 6th. A C6 has the notes C (1), E (3), G (5) and A (6). Sixth chords do NEVER contain a 7th! The major 6th can be found a whole step (2 frets) above the 5th. The 5th can be omitted.
Sixth chords have a dark, melancholic character. They create less tension than seventh chords and therefore don't tend that much towards a next chord change. Theoretically you can play a 6th chord instead or aditionally to all major, maj7 or V7 chords.
A C6 can also be seen as slash chord Am7/C. That is an Am7 chord with C in the bass („A minor seven over C”).