A 2-adjunction is a common name for various kinds of adjunctions in 2-category theory; not only adjunctions between 2-categories themselves, but more generally adjunctions within an arbitrary 3-category. Since there are various different levels of strictness at which one works in 2-category theory, there are various different kinds of 2-adjunction, including:
The word pseudoadjunction is sometimes used interchangeably with “biadjunction”, but often it refers specifically to a notion definable internal to a Gray-category. When specialized to the canonical Gray-category $Gray$, this produces a notion involving strict 2-categories and strict 2-functors, but pseudo natural transformations, and triangle identities up to isomorphism.
A lax 2-adjunction involves triangle identities only up to noninvertible transformation, and perhaps lax 2-functors and/or lax 2-natural transformations as well. If only the triangle identities are lax, this can be defined at any of the above levels of strictness and internalized in any of the above ways; but if the functors or transformations are lax, then it doesn’t fit very easily into a well-known abstract 3-categorical structure, since there is no 3-category (as usually understood) including lax functors or transformations.
There are also more specialized kinds of 2-adjunction, such as