First off, whenever you talk about nondeterminism you need to get rid of the idea of having an algorithm you execute to get a result. Nondeterministic models are descriptive only, not operational; there is no way to "execute" a nondeterministic algorithm. Sometimes, teachers say things like "the machine always guesses right" or "we execute all branches in parallel" but these intuition statements fall short in one way or the other.
So, accept that a nondeterministic machine describes some formal object. Period.
There are two ways to gain intuition about nondeterministic automata. From the lower end, consider finite-state transducers. They are essentially finite automata with output; obviously, FA reduce to them. In the non-deterministic case, each input can (but need not!) result in multiple outputs. Therefore, it makes sense to define the "result" of an FT $A$ on $w$ as the set of all outputs $A$ can produce on $w$. Now you can happily take the union over several input words, or consider preimages, and so on.
From the other end of the spectrum, consider NTM. The same idea works: for every input $w$, define as output the set of all tape contents you can have when the machine halts (in an accepting state) on $w$.
Note that nothing prevents you from requiring the automaton to have only one output per input, for instance when defining a complexity class.
It is similar for resource restrictions; the ideas used for defining decision problem classes should mostly carry over.
