Definition
Finite tensor category
A tensor category whose underlying linear abelian category is finite.
A finite tensor category over an algebraically closed field is a tensor category whose underlying -linear abelian category is finite abelian. In particular, its Hom spaces are finite-dimensional, its objects have finite length, it has finitely many simple isomorphism classes and enough projectives, and its monoidal structure is rigid with simple unit and -bilinear tensor product.
Why this is broader than fusion
A fusion category is precisely a semisimple finite tensor category under these conventions. Finite tensor categories may have nonsplit extensions of simple objects.
For example, over an algebraically closed field of characteristic , finite-dimensional representations of the cyclic group of order form a finite tensor category that is not semisimple. Its group algebra is , whose regular module has nontrivial extensions. This example is outside the often-adopted characteristic-zero convention for fusion categories.
References
P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories (2015), §4.1 and Chapter 6; author-hosted text.