The standard grips this chord is usually shown with.
Ebm6 is Eb, Gb, Bb and C, a minor sixth. The shape this page opens on is written xx1312, lowest string first.

Also written Ebmin6, Eb-6 or Eb minor 6. The same chord either way; which spelling you meet depends on who wrote the chart.
There are 8 common shapes for Ebm6 on the guitar, and 28 derived voicings in 3 families.
In which keys. Every note of Ebm6 belongs to C# major (the ii) and Bb minor (the iv).
The common shapes, written lowest string first: xx1312, x64546, x6857x, x68x78, xx8878, x-9-10-8-11-x, 11-x-10-11-11-11, 11-13-13-11-13-11.
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. Ebm6 has x645xx, 11-9-10-x-x-x, x6x57x, 11-x-10-11-x-x, x-6-10-11-x-x.
Drop 2. A close-position chord with the second voice from the top lowered an octave. Sits on four adjacent strings. Ebm6 has xx1312, 2313xx, x3434x, 6645xx, xx4546, xx8878, 8988xx, x-x-10-11-11-11.
Drop 3. The third voice from the top lowered an octave. The dropped voice becomes the bass, which is why a string is skipped. Ebm6 has 2x131x, x3x342, 6x454x, x6x576, 8x887x, x-9-x-8-11-8, 11-x-10-11-11-x, x-13-x-11-13-11.
Tensions. A shell of Ebm6, root, third and seventh, leaves a string free, and the tones that belong over it are the 9 (F) and the 11 (Ab).
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.