Am6 is built from A, C, E and F#. A is the root, C the minor third, E the fifth and F# the sixth. Said aloud, its notes are A, C, E and F sharp.
String by string, low to high, it plays A, E, A, C and F#. The root, A, shows up twice in the main shape. It runs from A on string 5 up to F# on string 1.
In Am6, only one string rings open, string 5, which gives A. String 6 stays silent. Your hand covers two frets, from 1 to 2. There is one more position, starting at fret 5.
Change only string 1 in Am6 and you have Am. Am6 and C have C and E in common. Moving to C, you keep two fingers down.
Am6 shows up in two keys. In G major it is the ii6 and in E minor the iv6. Further down there is a song that uses it.