First a reply to Jonathan, I guess you're right that the overturned Natural Spin Turn is kind of different from the regular or underturned ones. The technique book doesn't say much about overturning it besides you can do it and follow with say a Turning Lock to the Right. But I do see your point, overturning this figure does tend to change the move a bit more than the regular turn or underturning it. As for the Running Spin Turn, it looks like a kind of cool move (I saw it in the IDTA book), but with the syncopation, speed, and amount of turn, it's too hard for me to get right now.
To Andrew, figures that turn will normally have some standard amount of turn you're supposed to use according to the technique books. You can call this amount full, regular, standard or whatever you like. Underturning a figure means to do less turn (total) than the full amount and overturning it means to do more turn than the full amount. That's basically it. How much you underturn or overturn usually has certain amounts, too, but in practice if you're doing it for floorcraft purposes, you do whatever you need to in order to make things work.
For example, the full amount of turn on a Natural Spin Turn is 1-1/4 (1.25). Normally when you underturn it you turn 1 full turn and when you overturn it you turn 1-3/8. In all cases steps 1-3 have the same amount of turn, 3/8, and the difference is how much you turn on steps 4 (the pivot) and 5-6. The underturned Natural Spin Turn is easier to get at first, so I think most dancers are taught that version before learning it with the full amount and then how to overturn it. Your teacher should probably tell you that it's underturned though, just so you're aware of it. If you have other questions, feel free to ask, we all like discussing dancing. 
James Marshall
marshall@astro.umd.edu
http://www.astro.umd.edu/~marshall
