A pseudomonoid in the cartesian monoidal 2-category Cat is precisely a monoidal category. The general definition can be extracted from this special case in a straightforward way. The precise definition can be found in Section 3 of the paper of Day and Street referenced below.
Just as a monoid in a monoidal category can be equivalently defined as a monad in the corresponding one-object 2-category (the delooping of ), so a pseudomonoid in a monoidal 2-category can equivalently be defined as a pseudomonad in the corresponding one-object 3-category .
Ross Street and Brian Day, Monoidal Bicategories and Hopf Algebroids.