Question: So I've had at least two professors mention that backtracking makes an algorithm non-deterministic without giving too much explanation into why that is. I *think* I understand how this happens, but I have trouble putting it into words. Could somebody give me a concise explanation of the reason for this?

Answer: 
Consider an algorithm for coloring a map of the world. No color can be used on adjacent countries. The algorithm arbitrarily starts at a country and colors it an arbitrary color. So it moves along, coloring countries, changing the color on each step until, "uh oh", two adjacent countries have the same color. Well, now we have to backtrack, and make a new color choice. Now we aren't making a choice as a nondeterministic algorithm would, that's not possible for our deterministic computers. Instead, we are simulating the nondeterministic algorithm with backtracking. A nondeterministic algorithm would have made the right choice for every country.
Thought experiment:

1) Hidden from view there is some distribution of electric charges which you feel a force from and you measure the potential field they create. Tell me exactly the positions of all the charges.

2) Take some charges and arrange them. Tell me exactly the potential field they create.

Only the second question has a unique answer. This is the non-uniqueness of [vector fields](http://en.wikipedia.org/wiki/Vector_potential). This situation may be in analogy with some non-deterministic algorithms you are considering. Further consider in [math limits](http://en.wikipedia.org/wiki/Limit_of_a_function) which do not exist because they have different answers depending on which direction you approach a discontinuity from.