Smooth schemes play the role in algebraic geometry of manifolds in topology.
First, let X be an affine scheme of finite type over a field k. Equivalently, X has a closed immersion into affine space An over k for some natural number n. Then X is the closed subscheme defined by some equations g1 = 0, ..., gr = 0, where each gi is in the polynomial ring k[x1,..., xn].
In geometric terms, the matrix of derivatives (∂gi/∂xj) at a point p in X gives a linear map Fn → Fr, where F is the residue field of p. The kernel of this map is called the Zariski tangent space of X at p. Smoothness of X means that the dimension of the Zariski tangent space is equal to the dimension of X near each point; at a singular point, the Zariski tangent space would be bigger.
Define a variety over a field k to be an integral separated scheme of finite type over k. Then any smooth separated scheme of finite type over k is a finite disjoint union of smooth varieties over k. For a smooth variety X over the complex numbers, the space X(C) of complex points of X is a complex manifold, using the classical (Euclidean) topology.
For any scheme X that is locally of finite type over a field k, there is a coherent sheaf Ω1 of differentials on X.