A minor sixth is the chord you get when you take a minor triad and add the sixth note of the scale. They're notated Cm6.

For example, an F minor triad is F Ab C. The sixth note of the F scale (major or minor) is D, so the Fm6 chord consists of F Ab C D.

Compare with the major sixth: the only difference is the underlying triad.

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This chord is functionally similar to the minor seventh, flat 5 (m7b5) built on the relative minor: the component notes are identical, but the root note is of course different:

Gm6 = G Bb D E
Em7b5 = E G Bb D

The two can be interchanged with little bother for the sake of an effective bassline.