A category is additive if it satisfies four conditions, including existence of zero object, finite products and coproducts, and “coprod IMic to prod”. One proves that a category is additive iff Hom sets are abelian groups, composition is bilinear, and it admits finite products.
http://ncatlab.org/nlab/show/additive+and+abelian+categories
http://www.ncatlab.org/nlab/show/additive+envelope
nLab page on Additive category