The notes of G13 are G, B, D, F, A and E. G is the root, B the major third, D the fifth, F the minor seventh, A the ninth and E the thirteenth.
From the low string to the high one you hear G, F, B and E. This shape drops two chord tones, D and A. They are the fifth and ninth. The lowest note is G on string 6 and the highest is E on string 2.
G13 is a closed shape with no open strings. Move it to another fret and the chord changes, but the fingering stays. Strings 5 and 1 are not played. Everything fits between frets 3 and 5. The second shape on this page starts at fret 3.
From G13 to Gm7b5 only two strings change, 3 and 2. G13 holds the whole Em chord inside it, G, B and E. Going to Bm, one position stays the same.
G13 belongs to only one key, C major, where it is the V13.