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: 
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.
I like the maze analogy. Lets think of the maze, for simplicity, as a binary tree, in which there is only one path out. 

Now you want to try a depth first search to find the correct way out of the maze. 

A non deterministic computer would, at every branching point, duplicate/clone itself and run each further calculations in parallel. It is like as if the person in the maze would duplicate/clone himself (like in the movie Prestige) at each branching point and send one copy of himself into the left subbranch of the tree and the other copy of himself into the right subbranch of the tree.

The computers/persons who end up at a dead end they die (terminate without answer). 

Only one computer will survive (terminate with an answer), the one who gets out of the maze. 

The difference between backtracking and non-determinism is the following.

In the case of backtracking there is only one computer alive at any given moment, he does the traditional maze solving trick, simply marking his path with a chalk and when he gets to a dead end he just simply backtracks to a branching point whose sub branches he did not yet explore completely, just like in a depth first search. 

IN CONTRAST : 

A non deteministic computer can clone himself at every branching point and check for the way out by running paralell searches in the sub branches. 

So the backtracking algorithm simulates/emulates the cloning ability of the non-deterministic computer on a sequential/non-parallel/deterministic computer.