Complex Manifold
A complex manifold is a manifold with a complex structure, that is, an atlas of charts whose transition maps are all holomorphic.
Riemann Surface
A Riemann surface is a connected one‑dimensional complex manifold. Equivalently, it is a connected oriented real 2‑manifold endowed with a conformal structure that determines the same holomorphic coordinates in complex dimension one.
From the Riemann mapping theorem, we know that any simply connected Riemann surface is conformally equivalent to one of the three canonical Riemann surfaces: the Riemann sphere, the complex plane, or the unit disk.
Proposition
An almost complex structure is weaker than a complex structure: any complex manifold has an almost complex structure, but not every almost complex structure comes from a complex structure.