"radiation" is one of those words that says exactly what it means-- Something is emitting radiation, if something is radiating away from it. A powered light bulb emits photonic radiation. That's what it does.
There are many KINDS of radiation. Photonic, such as what comes out of a light bulb, or a laser light source (And also what comes out of radio broadcast towers. Most people dont really contemplate that radiowaves and light are basically the same thing, just the energy of the wavelengths involved differ.) There's also thermal radiation, and the various kinds of nuclear radiation. (Alpha particle emissions, Beta emissions, etc.)
Hawking radiation is another of those interesting kinds of radiation- It is high energy particle radiation, similar to nuclear radiation, except that it comes from some freaky quantum tunneling fun stuff. Basically, due to the uncertainty principle, you can never be precisely certain about the exact location of a quantum particle, nor its momentum or direction of travel. As such, particles appearing near an impermeable barrier, like the event horizon of a black hole, have a small but greater than zero probability of spontaneously being OUTSIDE the horizon when measured. Couple this with another facet of quantum physics madness-- "empty" spacetime is FAAAR from empty (It produces very short lived particle-like phenomena called vacuum fluctuations that exist as virtual particle pairs for the teeniest, tiniest amount of time possible before annihilating and returning to nothing. Really.) and you end up with a certain nonzero probability that these particles will come into existence near the horizon of the black hole, one part will fall in, and the other part will escape. This makes the energy fluctuation into a real particle pair, stealing energy from the black hole to do so, (shrinking it), and emitting that energy away from the horizon as a liberated high energy particle that tunneled its way past that impermeable barrier (the event horizon).
That's what hawking radiation is.