Spectre: Isn't the basic E chord shape by definition a power chord?
Good question. The answer is: No, it is not a "power chord" because it contains a 3rd, in this case the G# on the third string.

Guitar "power chords" are usually just the root and 5th. Sometimes they contain an octave duplication of the root.
So if you only played the bottom 2 or 3 strings of the basic E chord, then that would be a power chord.

Here's the movable shape with the root on the 6th string:

Here's the movable shape with the root on the 5th string:

Alternately they can be inverted, that is the 5th on the bottom and the root on top. (Think "Smoke on the Water").

Power chords are useful in Rock guitar playing for mainly two reasons:
- The ambiguity of having no third (which determines whether the harmony is major or minor) allows more freedom of choice for the soloist. Think about the bluesy history of rock with it's penchant to flat notes and bend them to "in-between" pitches and you'll understand how important this is.
- When guitarists started playing with increasingly more distorted tone, they found that playing the third in a chord made the sound more "messy" and that it didn't blend in the sonic mix of the ensemble.
Interestingly, two extremely distorted guitars playing in thirds works really well together, while a single guitarist playing the same intervals simultaneously will usually not find the results very satisfying. This effect can be heard in many harmonized guitar solos.
Here are a couple of examples excerpted from a Guitar Player magazine article, Harmony Guitar Solos of the '50s, '60s, and '70s:
Brian May’s astounding work with Queen frequently utilized counterpoint as well as parallel harmonies, often during the same song, and Ex. 11, excerpted from the eyebrow-raising entrance of May’s solo in “Keep Yourself Alive” (the opening track from 1973’s Queen), illustrates the point to a tee. He begins playing thirds over the tonic F and C (V) chords, and then switches to a counterpoint approach for the F-A7/E-Dm-Bb-C-F progression in the following two measures. The next four bars (not notated) find May adopting the same strategy—two bars of parallel harmonies followed by two bars of counterpoint. Bravo!
Our guided tour concludes with an excerpt from Thin Lizzy’s worldwide smash hit, “The Boys Are Back in Town” (from 1976’s Jailbreak). Ex. 12 shows how guitarists Scott Gorham and Brian Robertson coalesce during the songs’ infamous twin-guitar lead. Play through both parts, and you’ll find that the first three bars adhere to a rousing shuffle rhythm before contrasting with the dragged quarter-note triplets in bar 4. Play the whole deal over founder Phil Lynott’s A-B-C#-E bass line.

Have fun!