Quantum electrodynamics (QED) is the quantum field theory explaining how electrically charged particles interact with each other through the exchange of photons (light "quanta", or little packets of light). Photons, and therefore interactions in a QED, propagate at the speed of light. QED is referred to as a gauge theory, with a mathematically specified gauge field representing the electromagnetic force. The theory also explains magnetism, as magnetism and electricity are two manifestations of the same underlying force, electromagnetism.
The theory of QED is one of the most well-verified theories on Earth, sometimes giving precise results to ten decimal places, and was the first quantum field theory to be called consistent and complete. One prediction made by QED was found to be accurate up to .0038 parts per million, probably the most precise and accurate physical prediction ever made. Computing correct solutions to the behavior of systems with interacting parts or larger electron orbitals gets exponentially harder as the number of components increases, with some calculations requiring literally decades of work to compute and verify.
Out of the four forces of nature — electromagnetism, weak nuclear force, strong nuclear force and gravity — electromagnetism is probably the easiest to explain rigorously, although explaining it fully took many hundreds of scientists decades of work. The theory was developed to satisfaction in the late forties, thanks to the independent work of Sin-Itiro Tomonaga, Julian Schwinger and Richard Feynman. They received the 1965 Nobel Prize in Physics for their effort.
If electromagnetism were the only force of nature operating in the universe, QED would offer a complete account of its exact nature. However, it isn't, and the search continues for a quantum field theory which integrates all four forces. Furthermore, solving equations in QED is very difficult, more difficult than conventional quantum mechanics problems, as QED is a generalization of quantum mechanics to special relativity. The images most famously associated with QED are Richard Feynman's Feynman diagrams, which use straight and squiggly lines to analyze the different ways in which particles exchange photons to interact physically.
The theory of QED still produces mathematical infinities in certain contexts, and while many of these problems have been resolved, they persist at a certain level. Ad hoc renormalization algorithms have been developed to smooth over these theoretical imperfections. These infinities suggest that QED is not by any means a final theory, leaving the future open to the discovery of a more accurate theory, one which views electromagnetism in the context of the other three forces of nature.