The problem of calculating scattering cross sections in particle physics reduces to summing over the amplitudes of all possible intermediate states, in what is known as a perturbation expansion. These state can be represented by Feynman diagrams, which are much easier to keep track of in frequently torturous calculations. Feynman showed how to calculate diagram amplitudes using so-called Feynman rules, which can be derived from the system's underlying Lagrangian.
In addition to their value as a mathematical technology, Feynman diagrams provide deep physical insight to the nature of particle interactions. Particles interact in every way available; in fact, intermediate "virtual" particles are allowed to propagate faster than light. (This does not violate relativity for deep reasons; in fact, it helps preserve causality in a relativistic spacetime.) The probability of each outcome is then obtained by summing over all such possibilities. This is closely tied to the path integral formulation of quantum mechanics, also invented by Feynman.
The naive application of such calculations often produces diagrams whose amplitudes are infinite, pathological in a physical theory. The problem is that particle self-interactions are erroneously ignored. The technique of renormalization, pioneered by Feynman, Schwinger, and Tomonaga, compensates for this and eliminates the troublesome infinite terms.
Feynman diagram and path integral methods are also used in statistical mechanics.