Sixth chords, and the rest the seventh does not give
A sixth added to the triad, which sounds like rest where the major seventh would sound like colour; the ending chord par excellence in bossa and choro, and the chord charts confuse with the relative's minor seventh.
A major triad is rest, but it is too plain a rest to end a bossa nova. Add the major seventh and you gain colour, but the major seventh is a note a semitone from the root, and it does not fully settle. Add, instead, the sixth, and the chord rests with colour: that is the sixth chord, and it is the ending chord par excellence of half the Brazilian repertoire.
Four notes, none of them tense
C sixth has C, E, G and A. All four belong to the C major scale, none forms a semitone with the root, and there is no tritone. It is a chord that asks for nothing, which is why it serves as an ending: the music arrives at it and stops.
Compare with the major seventh chord — C, E, G and B. The B is a semitone from the C, and however beautiful it sounds, it keeps the chord slightly in the air.
Play both as the last chord of a phrase. The first ends beautifully; the second ends.
The sixth with a ninth
Bossa nova hardly ever stops on the sixth alone: it adds the ninth, and the chord of rest becomes C, E, G, A and D.
The slash here does not indicate a bass note, unlike the slash in nearly every other chart: "6/9" is the name of a quality, the sixth and the ninth together, and the bass is still the root. It is the final chord of countless recordings from the sixties, and the sound of "finished" on João Gilberto's guitar.
The minor sixth
The minor chord takes a sixth too, and there the sixth is major over the minor third: A minor sixth has A, C, E and F sharp.
The F sharp comes from the melodic minor scale, or from the Dorian mode, and it gives the minor chord a rest with a slight shine. It is the minor tonic chord of a lot of choro and a lot of jazz, and it is also a minor subdominant chord with an extra note, which makes it the favourite passing chord for anyone harmonising a descent.
The same chord, two names
C sixth has C, E, G, A. A minor seventh has A, C, E, G. They are the same four notes, and what decides which chord is which is the bass and the context: with C underneath, in a tonic position, it is C sixth; with A underneath, in a sixth-degree position, it is A minor seventh.
The chart chooses whichever helps the reading. In a cadence in C, writing C sixth says "rest"; writing A minor seventh says "sixth degree, on the way somewhere". Same sound, different reading, and the reading is what gets played.
What to take from here
The sixth chord adds the sixth to the triad, and the result rests with colour because none of its notes forms a semitone with the root or a tritone with another. With the ninth, it is bossa nova's final chord; in its minor form, it is a minor tonic with shine and a passing chord. The same four notes are the relative's minor seventh, and the bass and the context say which of the two you hear.