An additive category is
an Ab-enriched category (sometimes called a pre-additive category–this means that each hom-set is an abelian group and composition is bilinear)
which admits finite products (and hence finite coproducts).
In any Ab-enriched category, any finite product is also a coproduct, and dually. This includes the zero-ary case: any terminal object is also an initial object, hence a zero object (and dually). Such products are sometimes called biproducts and sometimes direct sums; they are absolute limits? for Ab-enrichment. (This does not extend to infinite products and coproducts.) In fact, an Ab-enriched category is Cauchy complete just when it is additive and moreover its idempotents split.
A pre-abelian category is an additive category which also has kernels and cokernels. Equivalently, it is an Ab-enriched category with all finite limits and colimits. An especially important sort of additive category is an abelian category, which is a pre-abelian one satisfying the extra exactness property that all monos are kernels and all epis are cokernels. See additive and abelian categories.
The Ab-enrichment of an additive category does not have to be given a priori. Any category with finite biproducts is automatically enriched over abelian monoids (as described at biproduct), so an additive category may be defined as a category with finite biproducts whose hom-monoids happen to be groups. (The requirement that the hom-monoids be groups can even be stated in elementary terms without discussing enrichment at all, but to do so is not very enlightening.) Note that the entire -enriched structure follows automatically for abelian categories.
Some authors use additive category to simply mean an Ab-enriched category, with no further assumptions.
The natural morphisms between additive categories are the additive functors.