basicly you need to know the C ionian mode for this
the number with the chord, is the step from the scale you need to add:
the scale is:
C D E F G A B C
1 2 3 4 5 6 7 8
so basicly
C6 would be adding the 6 to the major triad
which is c e g
im not explaing that good i guess, but maybe others will be able to explain better, or i'll take my time to do it better later today
by the way can anyone explain what is inverted chords or whatever the term is are about

later
by the way as you noticed numbers are between 2 and 13
basicly after 7, notes repeat themselves only in higher octaves, so the difference between 6 and 13 would be the octave you add the note from