Diminished chords can be produced simply by flatting the fifth found in a minor chord formula. Consequently, a C° or Cdim formula is R, ♭3, and ♭5 (C-E♭-G♭). Keep in mind that in a minor chord, the third is already flat, so only flatting the fifth is necessary (if you again flat the third, the chord would no longer be considered diminished).
Diminished seventh chords (°7th) are just traditional diminished chords with a double flat seventh attached. Their formula is R, ♭3, ♭5 and ♭♭7 or 6 (the double flat built from the root, flat third, flat fifth and minor (flat) seventh (ex. R, ♭3, ♭5, and ♭7).
Half diminished seventh chords or minor seventh flat five chords (m7♭5) are built from the root, flat third, flat fifth and minor (flat) seventh (ex. R, ♭3, ♭5, and ♭7).