The standard grips this chord is usually shown with.
E6 is E, G#, B and C#, a major sixth. The shape this page opens on is written 022120, lowest string first.
Note by note. E is the root, G# is the major third, B is the fifth and C# is the sixth.

Also written Emaj6, EM6 or E major 6. The same chord either way; which spelling you meet depends on who wrote the chart.
There are 11 common shapes for E6 on the guitar, and 30 derived voicings in 4 families.
One note away. Change a single note of E6 and you have E (with one note taken away), Emaj7 (with one note moved), Em6 (with one note moved), Eadd9 (with one note moved), E69 (with one note added) or E7 (with one note moved).
In which keys. Every note of E6 belongs to C# minor (the ♭III), E major (the I), F# minor (the ♭VII), Ab minor (the ♭VI), A major (the V) and B major (the IV).
The common shapes, written lowest string first: 022120, 022424, xx2424, x7665x, x79999, x-11-9-9-9-9, 12-11-11-9-9-9, 9-11-9-9-12-9, 12-x-11-13-12-x, 12-x-14-13-14-12, 12-14-x-13-14-12.
Under the hand. 2 of the 11 shapes here sit in open position, and the first uses 2 open strings. 6 of the 11 need a barre. Together they cover the neck from the 1st fret to the 14th.
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. E6 has 046xxx, x766xx, 12-11-11-x-x-x, x7x69x, 12-x-11-13-x-x.
Drop 2. A close-position chord with the second voice from the top lowered an octave. Sits on four adjacent strings. E6 has x2212x, xx2424, 4424xx, x4645x, xx6657, 7766xx, xx9999, 9-11-9-9-x-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. E6 has x2x120, 4x242x, x4x454, 7x665x, x7x697, x-11-x-9-12-9, 9x999x, 12-x-11-13-12-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. E6 has x7660x, 0x660x, xx6600.
Tensions. A shell of E6, root, third and seventh, leaves a string free, and the tones that belong over it are the 9 (F#) and the #11 (A#).
What kind of chord this is. E6 is a major chord — root, major third, fifth. major chords on every root, and what they are for.
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.