a complex manifold
A smooth manifold is a space patched together from pieces of R^n so that calculus makes sense on it. A complex manifold is the same idea, but the local pieces are open sets of complex space C^n instead, and the rule for gluing them is stricter: where two patches overlap, the change-of-coordinates map must be holomorphic — complex-differentiable — not merely smooth. Holomorphy is a far rarer and more rigid condition than smoothness (it forces a function's real and imaginary parts to obey the Cauchy-Riemann equations, hence to be harmonic and analytic), so a complex manifold carries a much tighter structure than the underlying real manifold of dimension 2n that it sits on top of.
Precisely, an n-dimensional complex manifold is a topological space M with an atlas of charts phi: U -> C^n whose transition maps phi_b composed with phi_a^{-1} are biholomorphic (holomorphic with holomorphic inverse). The complex dimension is n; the real dimension is 2n. Every complex manifold is canonically an oriented real 2n-manifold, but the converse fails badly: most even-dimensional smooth manifolds admit no complex structure at all, and even the question of which spheres do (only S^2 and possibly S^6, still open) is hard. The simplest examples are C^n itself, the complex projective space CP^n (the workhorse of the subject), and any smooth projective variety; one-dimensional complex manifolds are exactly the Riemann surfaces.
What makes the theory powerful is that holomorphic functions are scarce: on a compact complex manifold the only global holomorphic functions are constants (a maximum-principle phenomenon), so all the interesting structure lives in holomorphic sections of line bundles, in cohomology, and in the interplay of the complex structure with metrics. A common pitfall: a complex manifold is not just a real manifold of even dimension that happens to look like C^n in coordinates — the holomorphic compatibility of charts is genuine extra data, equivalent to an integrable almost-complex structure J, and most almost-complex structures are not integrable.
Complex projective space CP^1 is the Riemann sphere. Cover it with two charts, U_0 = {[1 : w]} with coordinate w and U_1 = {[z : 1]} with coordinate z. On the overlap (where neither coordinate is 0 or infinity) the transition is z = 1/w, which is holomorphic. So CP^1 is a 1-dimensional complex manifold — equivalently the 2-sphere S^2 with its standard complex structure.
The Riemann sphere CP^1 = S^2: two charts glued by the holomorphic map z = 1/w.
Real dimension 2n is necessary but nowhere near sufficient for a complex structure: the sphere S^4, for instance, admits no complex structure at all. 'Even-dimensional' does not imply 'complex'.