I'm currently reading "Harmony(third edition)" by Walter Piston. I'm at the part with interval inversion; I understand pretty much everything so far but Harmonic Inversion is confusing me a little bit. I'm wondering if anyone could help.

Here is what the text says:

"Harmonic inversion is quite different matter. In this procedure the names of the notes remain the same, but the lower of the two becomes the upper, or vise versa, with the consequence that there is usually a change in both the general and specific names of the interval."

So, basically, the higher pitched note is lowered an octave? Or the lower note is raised an octave depending on how you want to invert it...
this is how it works,the intervals are inverted,notes stay the same (like you said,either lowered or raised an octave) :
unison-------> octave
minor 2nd ----> major 7th
major 2nd----->minor 7th
minor 3rd---->major 6th
major 3rd---->minor 6th
perfect 4th---->perfect 5th

recognize a pattern here?
Intervals are a measure of distance.

If you play an A with an E seven semitones above the A you are playing a perfect fifth interval.

However if you raise the A an octave you will be playing the same note but the distance between the two notes is now only five semitones. A five semitone distance is a perfect fourth and sounds different than a perfect fifth.

This creates a different sound despite using the same two notes (E and A) notes. Basically A is a perfect fifth below E and a perfect fourth above it.