A multiplicity free fusion category is a fusion category in which all of the fusion coefficients are $0$ or $1$.

That is, a fusion category $\mathcal{C}$ is multiplicity free if for all simple objects $A,B$ we have that $A\otimes B$ is a direct sum of simple objects, each isomorphism class appearing with multiplicity at most $1$.

Equivalently, this means that the spaces $\text{Hom}(A\otimes B,C)$ and $\text{Hom}(C,A\otimes B)$ are at most one dimensional for all triples of simple objects $A,B,C$.