The nLab article on 2-monads says:
For example, ordinary (non-strict) monoidal categories are the strict algebras for a strict 2-monad $T_{MC}$ on $Cat$, but usually we care about pseudo, lax, and oplax monoidal functors rather than strict ones. Strict monoidal categories are the strict algebras for a different strict 2-monad $T_{StrMC}$ on $Cat$.
Where can I get a description of such two 2-monads? I am confused by the strictness in $T_{MC}$ which produces non necessarily strict monoidal categories.
The vague idea is that $TC$ is not the ordinary free monoidal category on $C$ made of strings in $C$, but instead is made of binary trees with leaves from $C$. This just means refining a string to remember the order of parentheses, and $TC$ has isomorphisms between the trees corresponding to different parenthesizations of the same string.
You can find some discussion of this stuff in Lack's 2-categories companion; abstractly, the idea is to take the "flexible" or "cofibrant" replacement of the monad $S$ for strict monoidal categories. Such a replacement $S'$ exists for any 2-monad, with the property that strict $S'$-algebras are equivalent to pseudo $S$-algebras. This is a theoretical justification for mostly ignoring pseudo algebras. An interesting remark special to this case is that $S$ is equivalent to $T$, which means their strict algebras are also equivalent. Composing, strict $S$-algebras are equivalent to pseudo ones, which is a fancy way of stating Maclane's theorem that every monoidal category can be strictified. This is emphatically not generalizable to arbitrary monads, as the failure of the coherence theorem for symmetric monoidal categories implies.