Recall the pattern in ordinary algebra:
In a monoidal category one can consider monoid or algebra objects.
In a symmetric monoidal category one can consider commutative monoid or algebra objects.
Accordingly in higher algebra:
In a monoidal (infinity,1)-category one can consider algebra in an (infinity,1)-category.
In a symmetric monoidal (infinity,1)-category one can consider commutative algebra objects.
A commutative algebra object in a symmetric monoidal (infinity,1)-category $C$ is a lax symmetric monoidal $(\infty,1)$-functor
In more detail, this means the following:
Given a symmetric monoidal (infinity,1)-category in its quasi-categorical incarnation as a coCartesian fibration of simplicial sets
a commutative algebra object in $C$ is a section
such that $A$ carries collapsing morphisms in $FinSet_*$ to coCartesian morphisms in $C^\otimes$.
the above definition is definition 1.19 in