The short version: electromagnetism is one of the four fundamental forces (the strong force, the weak force, and gravitation are the other three), responsible for all magnetic, electrical, and electromagnetic phenomena.
The long version is a little more complicated.
Start with history … phenomena we today call electrical have been known for millennia (e.g. static electricity), as have their magnetic counterparts (e.g. lodestone). The 17th and 18th centuries saw considerable scientific study of each, as separate forces, with Ørsted and Ampère uniting the two into electromagnetism, around 1820. Maxwell consolidated (in 1864) everything known about electromagnetism into what today we call Maxwell’s equations … and predicted electromagnetic waves (or radiation), a prediction verified by Hertz, two decades later. However, Maxwell’s equations opened a can of worms (to do with the aether, and the speed of light) … which lead to Einstein and special relativity. In parallel, a series of discoveries lead to photons (the quanta of electromagnetic radiation) and quantum mechanics, and these in turn to the recognition that the spectacular success of classical electromagnetism (i.e. Maxwell’s equations) actually depends on quantum field theory (with all its counter-intuitives).
Fast forward to the 1940s, and Quantum Electrodynamics (QED), which has electrically charged particles interacting via exchanges of photons (real or virtual), and describes all electromagnetic phenomena. QED is the most successful theory in physics, period (it has been tested, and found accurate, to one part in 1012!).
Here’s a fun fact: QED incorporates special relativity … and an electric charge (with no magnetic field) becomes an electric current (with an associated magnetic field), in relativity, simply by switching to a frame of reference moving with respect to the (stationary) electrical charge.
So, in its classical form, electromagnetism is an instantaneous ‘action at a distance’ type of force; in its quantum form; it’s an exchange of virtual photons, at the speed of light.
Now for more complication.
In 1979 Sheldon L. Glashow, Abdus Salam, and Steven Weinberg shared the Nobel Prize for Physics, for their contributions to the unification of electromagnetism and the weak force … which goes under the name electroweak interaction. So electromagnetism is just one manifestation of something more general, just as electricity and magnetism are two manifestations of one underlying thing, electromagnetism.
Want to learn more? Try Stargazers’ Electromagnetism, Math Pages on Maxwell’s equations, Richard Feynman’s excellent non-technical book on QED, and the 1979 Nobel Press Release on the electroweak interaction.
To get a handle on how diverse the roles of electromagnetism are, in astronomy, check out these Universe Today articles (just some of the many): Stellar Jets are Born Knotted,
Magnetic “Ropes” Connect the Northern Lights to the Solar Wind, and Spitzer Spies Ghostly Magnetar.
Astronomy Cast has an episode devoted to electromagnetism, called Electromagnetism. Some others you may also find interesting, on this topic, are The Search for the Theory of Everything, and The Important Numbers in the Universe.