In addition to Soujirou's writeup about inversions in music, there is a shorthand for writing inversions, and an additional inversion for every extension note added to the chord.
First, this shorthand involves the chord name followed by a subscript number. If the chord is G Major and it is in first inversion, then we can call it a G6, since the third on bottom and root on top makes a sixth, and the third to fifth makes a third, but that is implied, so we simply call it a G6 chord.
The Second inversion can be called a G64, because there is a sixth between the fifth on bottom and the third on top, and a fourth between the fifth on bottom and the root in the middle.
( As a side note, people often make the two numbers vertical instead of side by side.)
In Seventh chords, we get another inversion, because there is another note. Now, first inversions are shortened to 65, to imply the seventh now included... A seventh chord Second inversion is a 653, but shortened to 63. And the new Third inversion is a 643 chord, but shortened to 43.
There is also One new inversion for the Ninth chord, another for the Eleventh, and another for the Thirteenth chord. These may be talked about later.....