As the background to my music theory is more classical in nature, it used to puzzle me when I saw jazz chords like C9, B♭11 or F13. I mean, I knew what a 9th, 11th and 13th note were but I wondered why you'd call a note a 9th rather than a 2nd, or a 13th rather than a 6th and so on.
After all, when you talk about chord, you're normally talking about notes independent of octave. If you describe something as a C7 chord, you're not saying anything about whether the E and B♭ are in the same octave or not.
I can't remember when, but the breakthrough came when I realised that a 9th chord isn't just a major triad with the 2nd added, but one with the 2nd and 7th added, an 11th chord is one with the 4th and 7th added.
(just as an aside: the fact 2+7=9 and 4+7=11 here is an unrelated coincidence. An 11th is 4th+octave but due to the 1-based indexing used, you add 7 not 8)
Now yes, I've seen the theory books where they show a C9 as C+E+G+B♭+D and a C11 as C+E+G+B♭+D+F and a C13 as C+E+G+B♭+D+F+A but that really didn't help emphasize that it's the existence of the 7th that makes the the chord sound like (and be described as) a C9, C11 or C13 respectively instead of, say a Cadd2, Cadd4 or C6.
The 3rd and 7th are really the defining notes of a chord in Jazz, particularly comping on piano where you expect the bass to provide the root. So the final light went off when I saw the closing Jazz riff of Ben Folds Five's Underground notated. There were a bunch of triads that were marked as 13th chords. So, for example, the voicing E♭+A+D was marked as F13.
Note that that voicing has just the 3rd, 7th and 13th. The 13th is also a 6th but by calling the chord F13, it's making it clear the 7th is there as well which gives the chord a very different direction it wants to go. The 7th makes the whole chord want to resolve to a B♭, which gives the 13th/6th (the D) more of a suspended feel it doesn't have in an F6 chord.
I find not only the 13th chord a great substitute for a 7th now, especially when it's the dominant resolving to the tonic, but I also love the 7th+3rd+13th/6th way of voicing it too.
I know this is Jazz 101 but it was a breakthrough moment for me, anyway :-)
UPDATE (2012-01-01): interesting discussion on this post now on Hacker News