A representation of any categorical object (e.g. "groupoid" = "1-category with only isomoprhisms") is simply a (nice) functor from that object to the category of Vector Spaces. Then, as Chris points out, the abstract representation theory of groupoids essentially reduces to the representation theory of groups.

The story becomes much richer in the Lie category, because then you should ask for the representation to be smooth. The story has not been completely told, and even the parts that have been told I don't know well. For a hint at some of the interesting behavior, see http://front.math.ucdavis.edu/0810.0066. I don't know anything about the algebraic category, but I believe that there's interesting stuff there too (as far as I can tell, algebraic stacks are more complicated than smooth ones).

Hopefully someone who knows the literature better than I can say more.

A representation of any categorical object (e.g. "groupoid" = "1-category with only isomoprhisms") is simply a (nice) functor from that object to the category of Vector Spaces. Then, as Chris points out, the abstract representation theory of groupoids essentially reduces to the representation theory of groups.

The story becomes much richer in the Lie category, because then you should ask for the representation to be smooth. The story has not been completely told, and even the parts that have been told I don't know well. For a hint at some of the interesting behavior, see https://arxiv.org/abs/0810.0066. I don't know anything about the algebraic category, but I believe that there's interesting stuff there too (as far as I can tell, algebraic stacks are more complicated than smooth ones).

Hopefully someone who knows the literature better than I can say more.