Triads and 7th Chords
The basic structure in tonal harmony is the chord.  The foundation of any chord is the triad.

Triads consist of three pitches arranged vertically in thirds.  (This arrangement of thirds is referred to as tertian harmony.)  The thirds can be arranged in four different ways, resulting in triads of different qualities.

Seventh chords, extensions of the tertian triad, consist of four pitches arranged vertically in thirds.  The fourth pitch, a third above the fifth, will invariably be the interval of a 7th, of some sort, above the root of the chord.  (The "root" serves as the foundational pitch of the chord -- the pitch the chord is built on.)

When identifying 7th chords, there are two aspects of quality that need to be noted -- the quality of the triad alone, and the quality of the 7th in relation to the root.  Below are the most common 7th chords.

Inversions are the vertical arrangements of chord members resulting in a pitch other than the root appearing as the lowest sounding pitch.

The table below illustrates the figured bass symbols for root position and inverted triads and seventh chords.

Below are examples of how the inversions work on the staff.

7th Chords