Augmented chords are a composite of the major chord formula, containing the root, third, and raised fifth. (e.g. C+ formula is R, M3, and #5 (C-E-G#).
Augmented seventh chords, (such as C+7) are a variation of major seventh chords, in that they contain a raised fifth along with a major seventh interval (i.e.. R, M3, #5, and 7). Therefore, a C+7 would contain the notes C-1st, E-2nd, G# sharp 5th, and B – 7th.
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, b3, b5, and bb7 or 6 (the double flat built from the root, flat third, flat fifth, and minor (flat) seventh (ex. R, b3, b5, and b7).
Half diminished seventh chords or minor seventh flat five chords (m7b5) are built from the root, flat third, flat fifth, and minor (flat) seventh (ex. R, b3, b5, and b7).