Number of accepting path of a non deterministic automaton
This concept is called the ambiguity of the NFA.
Typically, there are 3 classes of ambiguity in this context: Bounded, polynomially bounded, and exponentially bounded.

Every NFA has at most an exponential number of runs on a given word (this is easy to see).

Interestingly, there is a simple syntactic characterization of polynomially bounded NFAs:

An NFA has a polynomial number of runs on a word $w$ iff for every state $q$, there is at most one cycle from $q$ to itself on every word $x\in \Sigma^*$. See this for details.

Testing for bounded ambiguity is PSPACE-complete. A good starting point is this paper