Preamble: An almost complex structure on a manifold $M$ is an endomorphism $J : TM \to TM$ such that $J^2 = -1$. An almost complex structure $J$ is said to be integrable if the Nijenhuis tensor, $N_J$, vanishes. $M$ admits an integrable almost complex structure iff $M$ is a complex manifold, i.e. $M$ has an atlas of charts to the open unit disc in $\mathbb{C}^n$ with holomorphic transition functions.
When are two (almost) complex structures considered equivalent?
For example, if $J$ is an (almost) complex structure, then $-J$ is also an (almost) complex structure. In what sense is this a new (almost) complex structure? Indeed, Michael Albanese's answer on this question shows that there can be quite a few (almost) complex structures on a manifold.
A seemingly natural way to define equivalence for complex structures is bilomorphicity - two complex structures are equivalent if the associated complex manifolds are biholomorphic. This seems like a reasonable thing to do, but doesn't work if you have an almost complex structure which is not integrable (since in that case there isn't an associated complex manifold).