A ring is a set equipped with two binary operations, addition and multiplication, where the addition makes the set a group, and the multiplication distributes over addition. A rig is nearly the same, but we don’t demand additive inverses. A 2-rig is a categorification of rigs, where we give a category a notion of addition and multiplication which obeys a distributive law. The thing which corresponds to addition is the existence of all small colimits, and the thing which corresponds to multiplication is a symmetric monoidal structure.
I think the word “2-rig” was first used in HDAIII.
A 2-rig is a symmetric monoidal cocomplet category for which the monoidal structure preserves small colimits in each argument.
$\mathsf{Set}$, the category of sets and functions is a 2-rig with its cartesian monoidal structure.
$\mathsf{Vect}$, the category of vector spaces (over some chosen field) and linear maps is a 2-rig with tensor product giving the monoidal structure.
John Baez, James Dolan?, Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes, Advances in Mathematics 135, pages 145-206, 1998.
Martin Brandenburg?, Bicategorical colimits of tensor categories, preprint, 2020. arXiv:2001.10123