Play Em6 one string at a time and you hear E, G, B and C#. E is the root, G the minor third, B the fifth and C# the sixth. Said aloud, its notes are E, G, B and C sharp.
String by string, low to high, it plays E, B, E, G, C# and E. Of the six strings you play, three give E, the note that names the chord. The lowest and highest notes are both E, two octaves apart, on strings 6 and 1.
The open strings add notes to Em6 too. String 6 gives E, string 3 gives G and string 1 gives E. Every finger presses at fret 2. There is one more position, starting at fret 7.
Change only string 2 in Em6 and you have Em. Em6 and G have G and B in common. Moving to C, you keep one finger down.
Em6 shows up in two keys. In D major it is the ii6 and in B minor the iv6.