Sixth chords
Build sixth chords and see how the bass note decides the chord's name.
One chord has two names, and the bass note is what chooses between them.
Adding a 6th
A sixth chord adds the major 6th to a triad. C6 is 1 3 5 6, C E G A. The minor sixth chord Cm6 is 1 ♭3 5 6, C E♭ G A: the 6th is major even in the minor 6th chord.
Two names for the same notes
C6 has the same notes as Am7 (A C E G). Which name applies depends on the bass note: with C in the bass it is C6; with A in the bass it is Am7. This is a clear case of the bass deciding the harmony (lesson 7.6). Likewise, Cm6 (C E♭ G A) has the same notes as Aø7 (A C E♭ G).
Where you hear them
Sixth chords are common as tonic chords in older pop, swing and bossa nova, where a major 7th would sound too sharp-edged. For the bass, the 6th is a frequent colour note in lines over a major chord: the “1–3–5–6” pattern of early rock and roll.
Play the pattern above over C (one bar), F and G. Then play A under the same C6 notes and hear it become Am7.