Interval inversion refers to moving the lowest note of an interval up by an octave.
Under inversion, perfect intervals remain perfect, major intervals become minor and the reverse, augmented intervals become diminished and the reverse. (Double diminished intervals become double augmented intervals, and the reverse.) Traditional interval names sum to nine: seconds become sevenths and the reverse, thirds become sixes and the reverse, and fourths become fifths and the reverse. Thus a perfect fourth becomes a perfect fifth, an augmented fourth becomes a diminished fifth, and a simple interval (that is, one that is narrower than an octave) and its inversion, when added together, will equal an octave.

Contrapuntal inversion requires that two melodies, having accompanied each other once, do it again with the melody that had been in the high voice now in the low, and vice versa. Also called “double counterpoint” (if two voices are involved) or “triple counterpoint” (if three), themes that can be developed in this way are said to involve themselves in “invertible counterpoint.” The action of changing the voices is called “textural inversion”.
Invertible counterpoint can occur at various intervals, usually the octave (8va), less often at the 10th or 12th. To calculate the interval of inversion, add the intervals by which each voice has moved and subtract one. For example: If motive A in the high voice moves down a 6th, and motive B in the low voice moves up a 5th, in such a way as to result in A and B having exchanged registers, then the two are in double counterpoint at the 10th ((6+5)–1 = 10).

Invertible counterpoint achieves its highest expression in the four canons of JS Bach’s Art of Fugue, with the first canon at the octave, the second canon at the 10th, the third canon at the 12th, and the fourth canon in augmentation and contrary motion. Other exemplars can be found in the fugues in G minor and B-flat major [external Shockwave movies] from Book II of Bach’s Well-Tempered Clavier, both of which contain invertible counterpoint at the octave, 10th, and 12th.

When applied to melodies, the inversion of a given melody is the melody turned upside-down.
There are at least two kinds of melodic Inversion – chromatic and diatonic.

Consider the C major scale (above): the C is a whole step up from the D, therefore, in chromatic inversion of the C major scale, we move down a whole step from C, which brings us to B flat. From the second note of our original series of notes (that is, the D), the major scale takes another whole step up to E, so in the inversion, we move a whole step down (from the B flat) which puts us at A flat. Since the next interval (from E to F) on the major scale is a half step, we must make a half step down from the A flat and we end up on G. We continue moving through the sequence of notes according to the (WS-WS-HS) WS (WS-WS-HS) pattern of the major scale, starting from the original note, except we move down rather than up. This is a chromatic inversion (below). This inversion can be characterized as an operation on the group of 12 semitones of Western music (which is isomorphic to the group Z/12).

The operation of chromatic inversion on the group Z/12 is I(x) = (12-x)mod12. We number the notes as follows: C=0, D flat=1, D=2, E flat=3, E=4, F=5, G flat=6, G=7, A flat=8, A=9, B flat=10, B=11. Then chromatic inversion takes 0 to 12mod12 = 0, therefore C goes to C. Then D goes to (12-2)mod12 = 10, so D goes to B flat. This is just another way of explaining the same inversion pictured above.

Diatonic inversion, as far as I can tell, inverts notes according to the C major scale rather than Z/12. Therefore the inversion about C is as pictured below.

Diatonic inversion about a note X doesn’t necessarily conserve note values (that is, C doesnt always go to C, D doesnt always go to D, and so on) but it does conserve the interval between X and inverted notes, relative to the C major scale.