I could use some help understanding the following in C++, particularly the difference between an operator and a function:
There is no huge difference between functions and operators. You can think of an using operator, e.g., 'a+b', as a shortcut to the function operator+(a,b) which is defined for the types of a and b. Of course, operators on primitive types (like integers) and a few other exceptions are not necessarily defined like this.
Thus, to answer a few of your specific questions:
Is user-defined operator+() a function or operator?
A function that implements an operator.
Can operator operate on operands at compile-time? Do they always operate at compile time?
Since it is a function, it operates at run time, but in some cases compiler optimizations can do work at compile time for certain operators. I'm not 100% sure why you're asking this, so perhaps there is something I'm not aware of here.
In C++ you can override what the symbols +, -, ==, etc. do when applied to class instances. By defining the "operator+" method in class A, you are telling the compiler what to do with code like:
A a, b, c;
c = a + b; // the + here actually calls a.operator+(b)
It's also a function or more precisely an instance method, in the sense that it's something that gets called.
EDIT: see also http://en.wikipedia.org/wiki/Operator_overloading and http://en.wikibooks.org/wiki/C++_Programming/Operators/Operator_Overloading