Slash-Chord. On the left side of the slash you have a chord, on the right side the lowest tone (bass). The C/F (say „C over F”) is a C major chord with a low F played in the bass.
The C major chord consists of the tones C (1), E (3) and G (5). The bass on the right of the slash can indeed, but does not necessarily be part of the chord on the left. In this case it is not.
Looking at the chord from the bass notes point of view, you will most likely interpret it as a Fmaj9 chord without the 3rd being played. Strictly analytic speaking Fmaj7sus2. Though syntactically right, you will hardly find that kind of noation anywhere. You'd rather write Fmaj7(no3) or Fmaj7(omit3). However, the slash chord interpretation C/F is still preferable here.
Example chord progressions:
Several major chords (the diatonic chords of the Ist, IVth and Vth degree of the F major scale) are played over a static „pedal bass”: FC/FBb/FC/F respectively FBb/FC/FBb/F
Chromatic progression of major chords over a pedal bass: Db/FC/FEb/FD/F
Delay of resolution (in a V-to-I-resolution the bass already goes to the I, while the chord stays on the V before resolving to a 6/9 chord): Bbmaj7 1sus9 C/FF6/9 or Gm9 113 C/FF6/9
(let the C/F ring for a while before resolving to the final F6/9 chord)
Two other pleasing progressions: C/FG/FEm7 Am7 C/FF/GC