Here is the classic "music theory" definition of an inverted chord:
For a C chord you have the following notes -
1 - C
3 - E
5 - G
So normally, a C chord goes C, E, G, starting on the 1st tone. If it starts on E, and goes E, G, C, you have a first inversion C triad. If it starts on G, for G, C, E, then you have a second inversion. This is acutually all pretty useless knowledge unless you're reading music or playing the piano, in which case, it becomes critical.
For the guitar, most of the chords that you'll play will be inversions inversions anyway. So for the second part of your question, that's where you would use such a thing . . . playing the good ol' guitar. But unless you're planning on learning another instrument or getting deep into music theory, you shouldn't have to worry about any of this too much.