The Wikipedia article is more technical than it should be, and for the reader in a hurry not all that well written. Here is a summary of the main points as best I understand them:
Complex manifolds are analogous to smooth complex algebraic varieties, not to the singular ones. But that discrepancy is surmountable, because you can also have complex analytic varieties which can have similar singularities. Then the first and most important relation is that every complex algebraic variety is a complex analytic variety. Every Zariski open set is analytically open; analytic gluing maps are more general than algebraic ones; and the allowed analytic charts are more general than the allowed algebraic charts. Also every algebraic morphism is an analytic morphism, so you get a morphism between categories.
But the connection is better than that because of the GAGA principle (globally analytic implies globally algebraic). My understanding of GAGA is very sketchy, but I think that the following is correct. Among other consequences of GAGA, a closed analytic subvariety of a proper (equivalent to compact) algebraic variety is algebraic. An analytic isomorphism between two proper algebraic varieties is algebraic. I would suppose that there is a similar principle for compact fibrations as well.
So, if you make compact analytic varieties algebraically, you can't escape from the algebraic class. All projective analytic varieties are algebraic, and in dimension 1 all compact curves are projective. Moreover, there are limited ways for a compact analytic manifold to avoid being projective, by Moishezon's theorem and Kodaira's theorem. In practice, then, most of the complex manifolds that people make are algebraic.
Contrast all this with real algebraic vs real analytic. It is still true that (the real points of) a smooth real algebraic variety is a real analytic manifold. More strongly than in the complex case, although it is highly non-trivial, every compact real analytic manifold is real algebraic. But the real algebraic structure is massively not unique, even for a circle, and that makes all the difference.