Let $i : H \to G$ be a subgroup of finite index. The transfer map is a special homomorphism $V : G^\mathrm{ab} \to H^\mathrm{ab}$. The usual ad hoc definition uses a set of representatives of $H$ in $G$ and then you have to check that it is independent from this choice and that it is a homomorphism at all. I think this definition is not enlightening at all (although it is, of course, useful for explicit calculations). A better one uses group homomology. Namely, for a $G$-module $A$ there is a natural transformation $A_G \to \mathrm{res}^{G}_{H} A_H$, $[a] \mapsto \sum_{Hg \in H/G} [ga]$, which extends to a natural transformation $H_\*(G;A) \to H_\*(H;\mathrm{res}^{G}_{H} A)$ (usually called corestriction or transfer). Now evaluate at $A = \mathbb{Z}$ and $* = 1$ to get $G^\mathrm{ab} \to H^\mathrm{ab}$. One can then calculate this map using the explicit isomorphisms and homotopy equivalences involved; but now you know by the general theory that it is a well-defined homomorphism.

I would like to know if there is an even more "abstract" definition. To be more precise: Is there a categorical characterization of the morphism $V$ within the category of abelian groups which only useses the morphism $i$ in the category of groups and the adjunction $\mathrm{Grp} {\longleftarrow \atop \longrightarrow} \mathrm{Ab}$?