Let $f: M^m \longrightarrow N^n$ with $m=n$. The degree of $f$ can be defined as $\deg(f) \in \mathbb{Z}$ such that $f_*\big([M]\big) = \deg(f)\cdot [N]$, where $[M]$ and $[N]$ denote the fundamental classes.
Here, we must have consider $m = n$. My question is: is there any generalization of that definition when $m \neq n$? If yes, could you give me a reference, please?
I know there are other equivalent definitions. If the generalization is with another kind, please let me know as well.
The correct analogue is the cohomology class $f^*[N]$. If you prefer the point-counting definition of degree, the correct analogue is $[f^{-1}(p)]$ for a generic point $p$. This is the Poincare dual of $f^*[N]$.
Neither of these are numbers, but they are the best you'll get.