Lyapunov exponents (not to be confused with Lyapunov functions) are gadgets that describe the exponential rate at which the trajectories of infinitesimally close initial conditions diverge from one another under a certain time evolution. They were first considered in the context of the qualitative theory of ODE's; now they are used in a variety of disciplines; in particular they are fundamental objects in smooth ergodic theory.
Let $M$ be a compact $C^\infty$ manifold, $f:M\to M$ be a $C^1$ diffeomorphism. Then the classical Oseledets' Theorem says that there is a Borel measurable $f$-invariant subset $M_0$ of $M$ such that at each point $x\in M_0$ there are unique numbers $\{\chi^1_x,\chi^2_x,...,\chi^{l_x}_x\}$ and the tangent space splits into $(f,Tf)$-invariant subspaces $T_x M = \bigoplus_{i=1}^{l_x} L^i_x$ in a unique manner such that for any Riemannian metric on $M$ we have
$$\forall v\in L^i_x\setminus0:\lim_{|n|\to\infty}\dfrac{\log|T_xf^n v|-\chi^i_xn}{|n|}=0.$$
Further, with respect to any Borel probability measure $\mu$ that is invariant under $f$, $\mu(M_0)=1$. Heuristically, this means that asymptotically the rate at which trajectories of points infinitesimally close to $x$ along the $L^i_x$ direction(s) diverge from the trajectory of $x$ is approximately $n\mapsto e^{\chi^i_x n}$, where $n$ corresponds to applying the diffeomorphism $f$ $n$ times.
Note that Oseledets' Theorem, even as stated here, is quite a general theorem. Accordingly there are analogous theorems compatible with different contexts, and consequently Lyapunov exponents are available in different contexts, e.g. random matrices, stochastic processes, smooth ergodic theory and entropy theory.
