Am6 would be a minor chord with the 6th scale degree included: ACEF#. This chord occurs naturally in the Dorian Mode and the Ascending Melodic Minor scale. Most commonly I find this chord associated with Dorian.
As you may have noticed in other circumstances, chords and other notes used within a piece of music do not have to stay within the key specified. This most commonly is accomplished with a Modulation (moving to another key), or some sort of borrowing/secondary function. Within the Jazz idiom, you often find that most Major7 chord are built off Lydian, with #11, and most Minor7 chords are built off Dorian, with 13. So you may also find that openness or darkness, depending on voicing, of the Minor6 chord may just be used as the overall sound used for minor chords within a piece, as opposed to only being used on a specific minor chord within the scale that would naturally have the major 6th degree.