Adding an additional third on top of a root position triad forms a seventh chord, as this new note will be a seventh above the root.
Although a seventh can be added to any chord, by far the most common seventh chord is V7, with ii7 in a distant second place.
In a minor key signature, the ii chord becomes a half-diminished seventh. (The chord is considered half-diminished because the interval from the fifth to the seventh is a major third. In a fully diminished seventh chord, all three intervals would be minor thirds.)
Note that the leading tone is raised in the example below to create a harmonic minor scale. This affects the quality of the III-chord, V-chord, and vii-chord.
A seventh is a tendency tone, meaning it leads to another pitch. Sevenths tend to push down, and so the seventh of a V-chord (scale degree 4) pushes down to scale degree 3. Since the V-chord also has a leading tone, which pushes up to the tonic, V7 leads very strongly back to the I-chord.
Do not confuse a seventh chord (i.e. a four-note chord) with a seven chord (i.e. a chord with scale degree 7 as the root.) Ordinals (second, third, fourth, etc.) denote intervals while integers (two, three, four, etc.) denote chord roots.
However, just to make things more confusing, it is common parlance to say the integer after the root to denote a seventh chord, as in V7 ("five seven") or ii°7 ("two diminished seven"). Nevertheless, both of these are seventh chords, but not seven chords.
Of course, vii°7 is both a seventh chord AND a seven chord.
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