manifolds and cobordisms
cobordism theory, Introduction
A surface is a space of dimension 2.
In differential geometry this means a 2-dimensional smooth manifold or something thereby parametrized?.
In complex analytic geometry this usually means a complex manifold of complex dimension 2.
Similarly and more generally, in algebraic geometry an algebraic surface is a variety of algebraic dimension 2.
analog for dimension 1: curve, algebraic curve