By the general philosophy of cohomology, cohomology is essentially derived $\operatorname{Hom}$ (i.e. $\operatorname{Ext}$), and homology should be derived tensor product (i.e. $\operatorname{Tor}$).
For example, given a topological space $X$, a commutative ring $K$, and $K_X$ the corresponding constant sheaf on $X$, then the sheaf cohomology of $X$ with coefficients in a sheaf of $K$-modules $\mathcal{V}$ can be defined as $$H^\bullet(X; \mathcal{V}) := \operatorname{Ext}^\bullet_{K_X}(K_X, \mathcal{V}).$$ This agrees with the more common definition of sheaf cohomology as right-derived global sections.
In contrast to sheaf cohomology, I haven't really seen sheaf homology discussed very much. Would it be correct to say that $$H_\bullet(X; \mathcal{V}) := \operatorname{Tor}_\bullet^{K_X}(K_X, \mathcal{V}),$$ or does the established definition of sheaf homology deviate from the usual pattern? Also, are there any common applications of sheaf homology?