To me, excellent as the others are, engelbrekt's is the most direct answer to your question. I.e.
1) Every projective plane curve is a compact Riemann surface, essentially because of the implicit function theorem.
2) Conversely, every compact Riemann surface [immerses as] a projective plane curve because it has [enough] non constant meromorphic functions which almost embed it in the plane. Also every meromorphic function is the pullback of a rational function in the plane. So all the analytic structure is induced from the algebraic structure.
In higher dimensions, complex projective algebraic varieties are a special subcategory of compact complex spaces, namely those that admit [holomorphic is sufficient] embeddings in projective space.
More precisely, an n dimensional compact complex variety has a field of meromorphic functions that has transcendence degree ≤ n, and projective algebraic varieties are a subcategory of those (Moishezon spaces) for which the transcendence degree is n. Indeed I believe Moishezon proved the latter are all birational modifications of projective varieties.
I would also add something about the impact of Riemann surfaces on algebraic geometry. Namely it was Riemann's introduction of the topological and analytic points of view, showing that path integrals and differential forms could be profitably used to study projective algebraic curves, that deepened and revolutionized algebraic geometry forever.
