A confusing name for a common chord. Dominant sevenths are the ones notated G7. Most musicians will just call them 'sevens', although this may be confusing, what with the diminished seventh, major seventh and minor seventh chords all lurking a few notes away.

To add to the confusion, dominant sevenths start on the dominant note of the scale -- eg, the fifth note. So if you're in a classical music exam and you're asked to play a dominant seventh in the key of C, then you'll have to start it on G.

For example's sake, here's the long-winded way to construct a Dominant Seventh in the key of F.
Dominant Sevenths begin on the fifth note of the scale. The fifth note of the F scale is C.
Construct a major triad on that note. A C major triad contains the notes C E G.
Add the seventh note up from the root note, in the base scale. The scale is F, the root note we have found is C. So, we have to count up seven notes in the F scale, starting on C. ...C D E F G A Bb...

The result is C E G Bb, otherwise known as C7, C dominant seven, or -- most properly -- the dominant seventh in the key of F.

But if you just want to work out, say, F#7, then it's F# major plus the note a tone below F#: F# A# C# E.

Functionally, this is traditionally a cadential chord resolving to the tonic (the unshakeability of this function in classical theory is where all the start-on-the-fifth-note malarkey came from.) The tonic chord may be major or minor; resolving to a major chord at the end of a piece in a minor key is a distinctive trait of Bach et al. called a Tierce de Picardie.

This cadential function has not been lost in Western music, and in fact remains very strong, but it is now no longer restricted to that function.