We can make our chords, and therefore our music, much more interesting if we realize that we do not have to use the 1st note of the scale on the bottom of every chord. We can change the order of the three scale notes (1st, 3rd and 5th) to get more variety in the sound of the chords we play. If we play the scale notes from lowest to highest pitchwise in the original order 1st, 3rd and 5th, the result is called a Root Position Triad.

C major chord in Root Position:
C (1) - E (3) - G (5)

If we leave off the bottom note and make the 3rd the lowest note pitchwise, then play the 5th higher and play the 1st an octave higher than originally played, the result is still the same chord, it is still a C major chord, but it is a different voicing of the C major chord. The result is called a First Inversion Triad.

C major chord in First Inversion:
E (3) - G (5) - C (1)

Let's do this again, but this time leave off two of the bottom notes, the 1st and the 3rd. Now the 5th is the lowest note pitchwise and the 1st and 3rd are both played an octave higher than originally played. Again, we have the same chord, but a different voicing. This results in what is called a Second Inversion Triad.

C major chord in Second Inversion:
G (5) - C (1) - E (3)

If we do this again we will move the lowest note, now the 5th, up an octave and we will be right back where we started at a Root Position Triad chord voicing. Except, now, the entire chord will be up an octave from the original location!

C major chord in Root Position:
C (1) - E (3) - G (5)

In this manner we can "cover the fretboard" with one single major chord!

The strict musical defintion of chord inversion is as follows:

A chord inversion is a chord in which the lowest note is not the root note. Therefore, a First Inversion Chord has the third of the chord as the lowest note (or "in the bass voice"). A Second Inversion Chord has the fifth of the chord as the lowest note (or "in the bass voice"). A Third Inversion Chord has the seventh of the chord (which of course must be a seventh chord of some kind) as the lowest note (or "in the bass voice").