To briefly answer the question in the title, the Toda lattice is a system of particles connected with springs having an exponential force law. It turns out that the exponential force law is very special, allowing one to completely describe the dynamics of the system. In this talk, we will solve the Toda lattice explicitly in the case of two particles, and describe the long-term dynamics of the system by analyzing the solution. Then we will show how the method can be generalized so that any number of particles can be handled with similar precision. The general method involves a surprising connection with linear algebra.