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A half-diminished seventh chord consists of a minor third, diminished fifth and minor seventh above the root - a full diminished seventh is the same but with (surprise!) a diminished seventh rather than a minor one. In major keys, vii7 is a half-diminished seventh and it is rare for it to resolve by fifth as in the first example below - the only common exception is as part of a chain of sevenths. More often vii7 in the major acts as a substitute for the dominant by resolving onto I. In older theory textbooks you might see this chord explained as a dominant ninth without a root, partly because of this tendency to resolve to the tonic as in the second example.
In minor keys, ii7 is a half-diminished seventh and this often resolves by fifth to V as part of a perfect cadence. As in the example below, the half-diminished seventh in this context is often in first inversion.
The half-diminished seventh is also used out of its diatonic context as a chromatic chord (see chromatic chords).
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