Functional analysis can be defined as the study of objects (and their homomorphisms) with an algebraic and a topological structure such that the algebraic operations are continuous. But usually the algebraic structure is fixed to be the one of a vector space.
Various sets of real or complex-valued functions (usually continuous or at least measurable) have not only the structure of a vector space but also an additional topological structure. To study these systematically, various classes of topological vector spaces were gradually developed and studied, often irrespective of the nature of the elements. Hence one can study (for example) Banach spaces of anything, not necessarily of functions. Naturally more and more general structures were studied.
Thus functional analysis is a field of mathematics studying compatible algebraic and topological structure, where ‘algebraic’ most often refers to linear spaces with structure (e.g. ordered vector spaces, real algebras etc.) and ‘topological’ may refer to mere topology but also to metric refinements like norm etc. The underlying ground field is most often real or complex numbers. In addition to the study of topological vector spaces, various interesting classes or examples of operators on them are in focus of this subject.
Main classical areas is the subject of topological vector spaces (important classes include Hilbert spaces, Banach spaces, Frechet spaces, and a pretty general class of locally convex topological vector spaces). The spectral theory, measure theory, ergodic theory and representation theory of these gave rise to the study of operator algebras, out of which the mainstream variety of noncommutative geometry also arose.
The English Wikipedia entry has a fair list of books on the subject.