C under a hand
Strummed from the lowest string to the highest, this grip sounds C, G, C, E, G and C, which are the degrees 1 5 1 3 5 1 rather than the 1 3 5 of the written chord. The root is the lowest note, so this grip is in root position and sounds the way the written chord suggests. With C in the bass and six strings sounding three notes, the 1 and 5 appear more than once.
Strummed from the lowest string to the highest, this grip sounds E, C, E, G, C and E, which are the degrees 3 1 3 5 1 3 rather than the 1 3 5 of the written chord. The lowest note is the 3 rather than the root, so this grip is in first inversion, written C/E. It has the same notes as the root-position grip, but with E in the bass it sounds different. With E in the bass and six strings sounding three notes, the 1 and 3 appear more than once.
Strummed from the lowest string to the highest, this grip sounds G, C, G, C, E and G, which are the degrees 5 1 5 1 3 5 rather than the 1 3 5 of the written chord. The lowest note is the 5 rather than the root, so this grip is in second inversion, written C/G. It has the same notes as the root-position grip, but with G in the bass it sounds different. With G in the bass and six strings sounding three notes, the 1 and 5 appear more than once.
Three shapes for C on guitar: C, C/E and C/G.
Every grip here is C, but no two of them sound alike. A chord with the root as its lowest note is in root position, and a chord with any other note at the bottom is an inversion. On a keyboard you choose the inversion you want, but on a neck the shape your hand can reach usually decides it for you. When a grip sounds a note more than once, the fretting hand decides which note that is, not the harmony.
An open chord uses open strings alongside fretted notes, and it has more strings than the chord has notes, so some notes are played twice. This is called doubling. The open C chord plays C, E, G, C and E: five strings sounding three note names, with the C and the E doubled.
The choice of which notes to double was not made for harmonic reasons. It follows from what the hand can reach while leaving the open strings ringing, which is why these shapes can look arbitrary until you name their notes. If you play one string at a time and say each note aloud, you will only ever hear three names.
Doubling also explains why open chords sound the way they do. Two Es an octave apart give a different sound from the same three notes played once each, and that difference comes from the instrument rather than from the theory.