I've run into a bit of an issue while reading through Farb and Margalit's "Primer on Mapping Class Groups," which I'm sure is due to my own incomplete knowledge! In chapter 6, they discuss the representation $\mbox{Mod}(S_g) \xrightarrow{\Psi} \mbox{Aut}(H_1(S_g;\mathbb{Z}))$. I have almost no knowledge of representation theory outside of the basic definitions, but isn't $H_1(S_g;\mathbb{Z})$ supposed to be a vector space? Everything would make perfect sense to me if we were working with field coefficients, but when our surface is something as basic as the 2-torus (for example), how could $\mathbb{Z}^2$ be a vector space over any field, let alone the "most natural" choices of $\mathbb{Q}$ or $\mathbb{Z}/p\mathbb{Z}$?
Any attempts to help me overcome my own incompetence are greatly appreciated!
You can very well have representations over arbitrary groups, or in fact any type of "structure": for any $X$ that has a reasonable $Aut(X)$, a representation of $G$ on $X$ is a morphism $G\to Aut(X)$.
A lot of the time in linear representations we work over fields because it's somewhat easier (vector spaces are easier to understand than modules, most of the time), but it's sometimes also interesting to work over $\mathbb Z$, that is, to work with abelian groups.