Play C6 one string at a time and you hear C, E, G and A. C is the root, E the major third, G the fifth and A the sixth.
From the low string to the high one you hear C, E, A, C and E. C rings twice across the five strings, and it is the root. G, the fifth, does not sound in this fingering. The lowest note is C on string 5 and the highest is E on string 1.
In C6, string 1 is played open and adds E. Avoid string 6. Everything fits between frets 1 and 3. The second shape on this page starts at fret 8.
C is one change away. String 3 is left open, bringing in G, the fifth of C. C6 holds the whole Am chord inside it, C, E and A. The change that keeps the most fingers still is to Am. Three stay in place.
C6 shows up in six keys. In C major it is the I6, in F major the V6 and in G major the IV6. The rest appear in the key list.