Fixed-point theorem is a result about existence of fixed points, i.e. points fulfilling F(x)=x, under some conditions on the function F.
Results of this type appear in many areas of mathematics, e.g. functional analysis (Banach), algebraic topology (Brouwer), lattice theory (Knaster-Tarski, Kleene) etc. They also have many important applications.
The importance of this area of mathematics is reflected in the large body of literature on the topic, we can mention famous monograph by Dugundji and Granas, which has almost 700 pages.