Originally Posted by: cmaxaSo are these octaves simply differentiated by their hertz measurement?[/quote]
An octave is an interval consisting of two notes, one of which is double the vibration frequency measured in hertz. However, octaves are also differentiated by where you can find them in order to play them on the instrument.

The lowest C on the guitar 130.81Hz & the next highest C is twice that at 261.63Hz.

Originally Posted by: cmaxa I guess what I'm struggling with is that we are supposed to count out the intervals when playing on one string, but I'm not seeing how that equates when we move two strings over and then down several frets to find the direct octave. It seems like we're just directed to do that, but I don't understand why quite yet.

Because you can often play the same note in more than one place on the guitar, these notes can be found & played in multiple locations.

So, it is valuable to know that you play the C major scale like this:

E ||---|---|---|---|---|---|---|---|---|---|---|---|
B ||---|---|---|---|---|---|---|---|---|---|---|---|
G ||---|-A-|---|-B-|-C-|---|---|---|---|---|---|---|
D ||---|-E-|-F-|---|-G-|---|---|---|---|---|---|---|
A ||---|---|-C-|---|-D-|---|---|---|---|---|---|---|
E ||---|---|---|---|---|---|---|---|---|---|---|---|

Or you can play the exact same notes but some of them in a slightly different location.

E ||---|---|---|---|---|---|---|---|---|---|---|---|
B ||---|---|---|---|---|---|---|---|---|---|---|---|
G ||---|---|---|-B-|-C-|---|---|---|---|---|---|---|
D ||---|---|-F-|---|-G-|---|-A-|---|---|---|---|---|
A ||---|---|-C-|---|-D-|---|-E-|---|---|---|---|---|
E ||---|---|---|---|---|---|---|---|---|---|---|---|

[QUOTE=cmaxa]So would the C on the high E string vibrate twice as fast as the C on the low e string?

You've got the right idea, but not quite the right application. As you correctly note, you can play the low C on the guitar in two different places.

E ||---|---|---|---|---|---|---|---|---|---|---|---|
B ||---|---|---|---|---|---|---|---|---|---|---|---|
G ||---|---|---|---|---|---|---|---|---|---|---|---|
D ||---|---|---|---|---|---|---|---|---|---|---|---|
A ||---|---|-C-|---|---|---|---|---|---|---|---|---|
E ||---|---|---|---|---|---|---|-C-|---|---|---|---|

Therefore if you play the C major scale starting on the low E string 8th fret, you could play the C major like this:

E ||---|---|---|---|---|---|---|---|---|---|---|---|
B ||---|---|---|---|---|---|---|---|---|---|---|---|
G ||---|---|---|---|---|---|---|---|---|---|---|---|
D ||---|---|---|---|---|---|-A-|---|-B-|-C-|---|---|
A ||---|---|---|---|---|---|-E-|-F-|---|-G-|---|---|
E ||---|---|---|---|---|---|---|-C-|---|-D-|---|---|

Which means the next C (the note an octave higher than your low C) is on the D string 10th fret. If you continue playing the C major scale from that C:

E ||---|---|---|---|---|---|-B-|-C-|---|---|---|---|
B ||---|---|---|---|---|---|---|-G-|---|-A-|---|---|
G ||---|---|---|---|---|---|-D-|---|-E-|-F-|---|---|
D ||---|---|---|---|---|---|---|---|---|-C-|---|---|
A ||---|---|---|---|---|---|---|---|---|---|---|---|
E ||---|---|---|---|---|---|---|---|---|---|---|---|

Then you get to the C on the high E string 8th fret, which is two octaves higher than the low C. Make sense? :)

Here are all three Cs with octaves indicated.

E ||---|---|---|---|---|---|---|-C2|---|---|---|---|
B ||---|---|---|---|---|---|---|---|---|---|---|---|
G ||---|---|---|---|---|---|---|---|---|---|---|---|
D ||---|---|---|---|---|---|---|---|---|-C1|---|---|
A ||---|---|---|---|---|---|---|---|---|---|---|---|
E ||---|---|---|---|---|---|---|-C-|---|---|---|---|

Lowest C = C
1 octave higher = C1
2 octaves higher = C2

The end goal of all this info is to be able to know how it is possible to manipulate the guitar in order play notes. Eventually to play music. :) Keep going forward in GF2. This will all make more sense as you get into scales.