An $$n$$-dimensional complex manifold is a topological manifold equipped with an open cover $$\\{U_{\alpha}\\}$$ and homeomorphisms $$\varphi_{\alpha} : U_{\alpha} \to B$$ where $$B$$ is an open ball in $$\mathbb{C}^n$$ such that the transition maps $$\varphi_{\alpha}\circ\varphi_{\beta}^{-1}$$ are holomorphic.
Every complex manifold is smooth, but not every smooth manifold is complex. An intermediate notion is that of an almost complex manifold which is a smooth manifold $$M$$ equipped with a bundle endomorphism $$J : TM \to TM$$ such that $$J\circ J = -\operatorname{id}_{TM}$$. Such a $$J$$ is called an almost complex structure.