Quantum electrodynamics, abbreviated QED, is a relativistic quantum field theory that arises when we apply the principles of quantum mechanics to electromagnetism and electrodynamics. QED covers every possible interaction between an electron (or a positron) and a photon.
A hand-wavy way to picture how QED works is to imagine electromagnetic fields reduced to a grid, and then the forces on electrically charged particles are described in terms of the exchange of photons between the particles (the photon is the carrier of the electromagnetic force). While the mathematics of QED, like all quantum field theories, are pretty esoteric, interactions involving QED can be conveniently and relatively painlessly understood through the use of Feynman diagrams, which look like the sort of things you scribble while talking on the phone.
When quantum mechanics came around in the 1920s and 1930s, physicists realized that they ought to try to apply it to all other areas of physics, including electrodynamics. QED was developed in the 1940s, largely by Richard Feynman, Julian Schwinger, Sin-Itiro Tomonaga, and Freeman Dyson. Feynman, Schwinger, and Tomonaga received a Nobel Prize in Physics for their work.
Relativistic quantum field theory is hard. Even the fundies and nutjobs (well...most anyway) stay away from stuff like this for the most part and mostly stick to misinterpreting regular ol' quantum mechanics if they want to take on small scales. If you want to learn more, get outside the dollhouse.