Ok, there are two separate answers here:
1) In astrophysics, there are some situations where black holes seem to emit lots of radiation, which we can observe. What is really happening here is that a lot of stuff (gases and dust from other stars) is falling into the black hole. Before the stuff gets to the black hole, it gets heated up (basically by friction of one sort or another, that's a complicated story that isn't fully understood) and hot stuff emits light that we see. This doesn't go into the black hole because it is aimed outwards. This is just like if you have a comet near the earth, it might crash into the earth, or it might just go around it, miss, and head off into space. That is the case of light from a black hole that we observe.
2) There is also the idea that black holes "evaporate" and actually that part of the black hole leaves the black hole and radiates away. To our knowledge, scientists have never seen this. The radiation would be composed of every possible kind of particle to some degree, but I would think mostly particles of light (photons) because they have no rest mass and would be least affected by the black hole. The idea is that, in quantum mechanics, the physics of fundamental particles, particles can actually jump through energy barriers higher than they should be able to (for example, like if you were able to teleport straight through a hill instead of having to walk over it). So particles inside the black hole can "jump" out (it is called "tunneling") and then escape the black hole -- if they are aimed straight out and moving very fast, like the speed of light, they can get away. So this is black hole evaporation. Actually, this is a very oversimplified picture; really, it is the bending of space near the black hole that creates the particles which escape away -- along with negative energy particles that fall into the black hole and reduce its energy. But how that happens is very complicated. Black hole evaporation is also not completely understood, but many scientists are studying it using something called "string thoery" which tries to understand gravity and quantum mechanics all at once by imagining fundamental particles are really little bits of 1-dimensional string like objects.
I believe the scenario for evaporating black holes is the following:
At the event horizon of a black hole an electron-positron pair is created out of the vacuum due to quantum fluctuations.Sometimes, the electron might shoot away from the black hole, while the positron goes in the opposite direction into the black hole. When this happens, the black hole has lost a little mass. Maybe the positron shoots out away from the black hole instead, hits an electron and they annihilate, emitting photons (radiation) in the process. Since all this occurs just outside the event horizon, any light generated can escape to be seen by us.
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