The standard grips this chord is usually shown with.
C6 is C, E, G and A, a major sixth. The shape this page opens on is written x32210, lowest string first.

Also written Cmaj6, CM6 or C major 6. The same chord either way; which spelling you meet depends on who wrote the chart.
There are 10 common shapes for C6 on the guitar, and 36 derived voicings in 4 families.
In which keys. Every note of C6 belongs to C major (the I), D minor (the ♭VII), E minor (the ♭VI), F major (the V), G major (the IV) and A minor (the ♭III).
The common shapes, written lowest string first: x32210, x35555, x75555, 877555, 575585, 8x798x, 8-x-10-9-10-8, 8-10-x-9-10-8, x-x-10-12-10-12, x-15-14-14-13-x.
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. C6 has x322xx, 877xxx, x-15-14-14-x-x, x3x25x, 8x79xx.
Drop 2. A close-position chord with the second voice from the top lowered an octave. Sits on four adjacent strings. C6 has x0201x, xx2213, 3322xx, xx5555, 5755xx, xx7988, x-10-10-9-10-x, x-x-10-12-10-12.
Drop 3. The third voice from the top lowered an octave. The dropped voice becomes the bass, which is why a string is skipped. C6 has 3x221x, x0x010, x3x253, x7x585, 5x555x, 8x798x, x-10-x-9-10-8, 12-x-10-12-10-x.
Open string. An open string left ringing against a fretted grip. Because it costs no finger, it sounds a note the closed shape has none left for -- a tension on top, or a pedal root underneath. Either a familiar grip with open strings added, or a shape so sparse the open strings carry the chord. These belong to their key rather than to a shape, so D, A, G and E have many and Eb a handful. C6 has 030253, 00007x, x00070, 0x0075, 5x0070, 00403x.
Tensions. A shell of C6, root, third and seventh, leaves a string free, and the tones that belong over it are the 9 (D) and the #11 (F#).
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.