Let $f:M\to N$ be a continuous map between two oriented manifolds of dimension $n$. There are a lot of alternative definitions of degree, but I'm only interested in the following one. Since $M,N$ are oriented, that means that $H_n(M)=H_n(N)=\mathbb{Z}.$ So the induced map in homology $$ f_\ast:H_n(M)\cong \mathbb{Z}\xrightarrow{d}\mathbb{Z}\cong H_n(N) $$ is multiplication by an integer $d$, which we call the degree.
Next, I would like to generalize this definition to manifolds that are not oriented. There's an obvious way to do this. Let $p\in \mathbb{Z}$ be a prime, and let $f:M\to N$ be a map between manifolds such that $H_n(M)= H_n(N)= \mathbb{Z}/p\mathbb{Z}.$ Then $$ f_\ast:H_n(M)\cong \mathbb{Z}/p\mathbb{Z}\xrightarrow{[d]}\mathbb{Z}/p\mathbb{Z}\cong H_n(N) $$ is multiplication by some $[d]\in \mathbb{Z}/p\mathbb{Z},$ which I'm calling mod p degree.
This seems like such an immediately useful idea that somebody has to have done it before. But I don't know where in the literature to look. Does anybody have any suggestions?
Edit: Cross-posted to Math Overflow here.
This is only interesting for $p=2$, and this is indeed the mod-2 degree. Searching for this should give you many references.
For a map $f:M\rightarrow N$ between non-orientable manifolds you can sometimes refine the mod-2 degree and orient the map $f$ instead of the manifolds. You get a notion of twisted degree. I think this is due to Paul Olum, perhaps in the paper below.