The notes of C7 are C, E, G and Bb. C is the root, E the major third, G the fifth and Bb the minor seventh. Said aloud, its notes are C, E, G and B flat.
String by string, low to high, it plays C, E, Bb, C and E. The root, C, shows up twice in the main shape. This shape leaves out G, the fifth. It runs from C on string 5 up to E on string 1.
In C7, only one string rings open, string 1, which gives E. Avoid string 6. Your hand covers three frets, from 1 to 3. There is one more position, starting at fret 8.
Change only string 3 in C7 and you have C. C7 and Am have C and E in common. Moving to Am, you keep two fingers down.
You will find C7 in two different keys. In F major it is the V7 and in F minor the V7. Further down there is a song that uses it.