It is surprising that fixed point theorems (FPTs) appear in so many different contexts throughout Mathematics: Applying Kakutani's FPT earned Nash a Nobel prize; I am aware of some uses in logic; and of course everyone should know Picard's Theorem in ODEs. There are also results about local and global structure OF the fixed points themselves, and quite some famous conjectures (also labeled FPT for the purpose of this question).
Many results are so far removed from my field that I am sure there are plenty of FPTs out there that I have never encountered. I know of several, and will post later if you do not beat me to them 🙂
Community wiki rules apply. One FPT per answer, preferably with an inspiring list of interesting applications.
Best Answer
The Lefschetz Fixed Point Theorem is wonderful. It generalizes the Fixed Point Theorem of Brouwer, and is an indispensable tool in topological analysis of dynamical systems.
The weakest form goes like this. For any continuous function $f:X \to X$ from a triangulable space $X$ to itself, let $H_\ast f:H_\ast X\to H_\ast X$ denote the induced endomorphism of the Rational homology groups. If the alternating sum (over dimension) of the traces
$$\Lambda(f) := \sum_{d \in \mathbb{N}}(-1)^d\text{ Tr}(H_df)$$
is non-zero, then $f$ has a fixed point! Since everything is defined in terms of homology, which is a homotopy invariant, one gets to add "for free" the conclusion that any other self-map of $X$ homotopic to $f$ also has a fixed point.
When $f$ is the identity map, $\Lambda(f)$ equals the Euler characteristic of $X$.
Update: Here is a lively document written by James Heitsch as a tribute to Raoul Bott. Along with an outline of the standard proof of the LFPT, you can find a large list of interesting applications.