Analytic spaces are spaces that are locally modeled on formal duals of sub-algebras of power series algebras on elements with certain convergence properties with respect to given seminorms. This is in contrast to (formal) algebraic spaces ((formal) schemes) where no convergence properties are considered.
In complex analytic geometry analytic spaces – complex analytic space – are a vast generalization of complex analytic manifolds and are usually treated in the formalism of locally ringed spaces. In this case the GAGA-principle closely relates complex analytic geometry with algebraic geometry over the complex numbers.
In the case of non-archimedean ground field, the topology of the affine space is totally disconnected what requires different approach than, say, over complex numbers. This leads to several variants like rigid analytic geometry, Berkovich spaces. Huber's adic spaces and so on.