A half-diminished seventh chord is composed of four notes where: 1) there are three semitones between the first and the second note; 2) there are three semitones between the second and the third note; and 3) there are four semitones between the third and the fourth note.
The A half-diminished seventh chord for example is composed of A, C, Eb, and G. There are three semitones between A and C, three semitones between C and Eb, and four semitones between Eb and G.
The half-diminished seventh chord thus is denoted by "m7b5" (i.e., a minor seventh chord with a diminished fifth). The chord in the example above is Am7b5 = A, C, Eb, G.
A half-diminished seventh chord in other words is simply a minor seventh chord with a diminished fifth. A minor seventh chord would be composed of, say, the root of a scale, a minor third, a perfect fifth, and a minor seventh (i.e., notes that are three, six, and ten semitones away from the root respectively). The minor seventh chord Amin7 is composed of A, C, E, and G. Changing the perfect fifth to a diminished fifth (E to Eb) changes the minor seventh chord to a half-diminished seventh chord.
The first three notes of the half-diminished chord (A, C, Eb in the example above) form a diminished chord. We can then also say that that the half-diminished seventh chord is composed of a diminished chord and a minor seventh. If the minor seventh is changed to a diminished seventh, the chord becomes a diminished seventh chord (A*7 = A, C, Eb, Gb in the example above).
The half-diminished seventh chord occurs naturally in many scales. The following are examples.