The standard grips this chord is usually shown with.
Bbm6 is Bb, Db, F and G, a minor sixth. The shape this page opens on is written x13x23, lowest string first.

Also written Bbmin6, Bb-6 or Bb minor 6. The same chord either way; which spelling you meet depends on who wrote the chart.
There are 9 common shapes for Bbm6 on the guitar, and 31 derived voicings in 3 families.
In which keys. Every note of Bbm6 belongs to F minor (the iv) and Ab major (the ii).
The common shapes, written lowest string first: x13x23, xx3323, x4536x, 6x5666, 688686, x-x-8-10-8-9, x-13-11-12-11-13, x-13-15-12-14-x, x-x-15-15-14-15.
Shell. Root, third and seventh: the guide tones fix the quality, so the fifth is dropped. That is what leaves a string free, so pick a tension to put on top. Bbm6 has 645xxx, x-13-11-12-x-x, x1x02x, 6x56xx, x-13-x-12-14-x, x156xx, 6-10-11-x-x-x.
Drop 2. A close-position chord with the second voice from the top lowered an octave. Sits on four adjacent strings. Bbm6 has x1302x, xx3323, 3433xx, xx5666, x8868x, x-x-8-10-8-9, 9-10-8-10-x-x, x-10-11-10-11-x.
Drop 3. The third voice from the top lowered an octave. The dropped voice becomes the bass, which is why a string is skipped. Bbm6 has x1x021, 3x332x, x4x363, 6x566x, x8x686, 9-x-8-10-8-x, x-10-x-10-11-9, 13-x-11-12-11-x.
Tensions. A shell of Bbm6, root, third and seventh, leaves a string free, and the tones that belong over it are the 9 (C) and the 11 (Eb).
The common shapes and the derived voicings come from different places, and both are explained in where the chords and the theory come from.
Shows each chord the way this style would voice it, and marks which of its alternatives belong to the idiom.