Chord inversions are "spellings" of a chord with a note other than the fundamental in the bass. Let's take a simple example:
The C major triad consists of the notes C, E and G, in that order:
As long as the C is on the bottom, the chord is in root position. Both the above spelling (C E G) and this one (C G E):
... are in root position.
If we put the E on the bottom, the chord is in first inversion:
E-G-C and E-C-G are both the first inversion of the C major triad, because the E is on the bottom:
If the third note of the chord, in this case G, is on the bottom, the chord is in second inversion:
The chords can be in closed position:
... or open position:
... but that doesn't affect chord inversions. The only thing that matters is which note is in the bass.
Remember, then, that inverting a chord simply means changing the order of the notes so that a note other than the root of the chord is in the bass.
It's important not to confuse root and bass. The root is the note on which the chord is built, while the bass is the bottom note of the chord—even if it is not the root.
For example, in a C major chord, C is the root, whether or not it is the bass note. If the C major chord is in first inversion, then E is the bass note (i.e., the bottom), but C is still the root of the chord. It's still a C major chord, after all, so C will always be the root of that chord whether or not it's inverted.
A chord has as many possible positions as it has notes. Since our triad above has three notes, there are three possible positions: root position, first inversion and second inversion.
Since root position is not itself an inversion, a triad has two possible inversions. We can extrapolate this rule thus: The number of possible inversions of a chord equals the number of notes minus one.
Let's now turn to four-note chords and take a look at a very common chord, the dominant seventh. Here is the dominant seventh chord in C major, in root position:
Note that the dominant seventh is built on the fifth scale degree of any key. The fifth note of the C major scale (i.e., the fifth scale degree) is G.
The notes of this chord, from bottom to top, are G, B, D, F.
Now let's take a look at the chord inversions. If B (the second note of the chord) is on the bottom, this chord is in first inversion:
If the third note, D, is the bass note, our seventh chord is in second inversion:
Finally, if the fourth note, F, is in the bass, the dominant seventh chord is in third inversion:
Note that when determining the inversion, we're only concerned with the bottom note—the remaining notes can be in any order. It's therefore the bass note that determines the inversion.
This concludes our lesson on chord inversions. Practice hard and you'll soon be an expert!