An electromagnetic field is an extremely important concept in fields such as physics or electrical engineering, but despite its pervasiveness avoids easy definition. Since this is the electromagnetic field node, though, I'll take a crack at it. An electromagnetic field is a field of electromagnetic force created when any charged particle, most commonly an electron, experiences a change in velocity. Of course, changing currents, which produce electrical fields, as in an AC (alternating current) power sources, also have the potential to create electromagnetic fields- electromagnetism is a two-way street.
Electromagnetic fields are actually a two for one deal: there are both electric and magnetic forces at work within the field, each with their own effects on matter and energy passing through the field. These fields are called E-fields (for Electricity) and the H-field (the magnetic portion of the field with no convenient letter in common).
This is the easy one since you can plainly see the effects of it on ordinary physical objects as opposed to obscure quantum models, which should burn for all eternity. Did you ever have one of those toys where you use a magnet and iron filings to give the guy a moustache (or poofy bouffant hair and sideburns to mock your elders' yearbook pictures)? You just demonstrated the effects of an H-field on an object passing through its area. If you drop the magnet (which is probably a ferric dipolar bar magnet) onto the toy so it lays flat, you can see that the magnet pulls the filings into a characteristic pattern, kind of a double ellipse on either side. These are called lines of force. No, they're not actually there, but they're a convenient way to visualize what an H-field looks like and how they interact with such fun things as electrical fields and conductors. The lines, named by Faraday, show the direction and magnitude of the field at a certain point in space.
The field we just played with is an example of an electrostatic field- that is, it ain't moving or otherwise doing work. All permanent magnets have a constant electrostatic field that stays bound to the origin at all times, though it can be affected by other fields. The strength of this field (the force that it can exert) is measured in Gausses, and some types (such as rare-earth magnets) are more powerful than others (such as garden-variety ceramic or ferric magnets) even before other fields or types of energy begins to screw around with them.
Well, speak of the devil. Recognizable E-fields in the real world usually take the form of an ominous hum around transformers or the static your radio picks up when you move it near any big source of electricity (like the aforementioned transformer). This effect doesn't have to be specifically because of current, though- other forms of electromagnetic energy, such as microwaves, will wreak havoc on electronics of any sort because of the principle of mutual inductance. This is also why they used to ask you to turn off your cell phones on planes- there's been no real laboratory evidence to suggest the signals thrown out by cell phones overlap with, say, vital communications equipment, but the airlines claim anecdotal evidence and pout. Ask Boeing or Airbus about the possibility of electromagnetic interference with vital systems, though, and they produce volumes of research where they bombarded their planes with electromagnetism and...nothing happened. Oh well. Back to E-fields.
The electric field is created by the presence of an electric charge like an electron, ion, or proton. When the field is created, it will exert influence over all other charges within its area, just like a magnetic field.
Putting It Together
The two-way street relationship discussed above is actually symptomatic of a much more elegant solution to linking up electricity and magnetism: they're two aspects of the same force, hence the umbrella name electromagnetism. Since electromagnetism was originally dealt with as two separate forces, we divide the influences of the electromagnetic field into two halves, the H-field and the E-field, but this is mostly for ease of conception than any true separation of their characteristics and effects.
Applications
MRI. Flux compression generators. Computers. Cell phones. Palm pilots. Radios. GPS. Railguns. If it requires batteries or is plugged into a wall, chances are an electromagnetic field is somehow involved.
DISCLAIMER: This has been a discussion of classical electromagnetism. I don't know squat about relativistic electrodynamics or (horror of horrors) quantum electrodynamics, so I'll leave that to someone with a more than passing knowledge of basic physics.
You may also note that I have carefully avoided Maxwell's equations, Ampere's formula, or basically any math at all. This is because (1) they're too complicated for my tiny mind and (2) I don't think they're really necessary in a basic information node. Of course, if somebody feels the need to toss 'em in there...good. Then you can explain it to me.
SOURCES:
Frank, Nathaniel H., and John C. Slater. Electromagnetism. New York: McGraw-Hill
Book Company, 1947.
http://en.wikipedia.org/wiki/Electromagnetic_field
Pollack, Gerald L., and Daniel R. Stump. Electromagnetism. San Francisco: Addison
Wesley, 2002.
Suydam, Vernon A. Fundamentals of electricity and electromagnetism. New York: D.
Van Nostrand Company Inc., 1940.
http://www.eee.strath.ac.uk/ug-info/19222/lectures_notes2003/Basic%20Electromagnetism1.pdf