It was discovered that Black Holes were somehow radiating some form of electromagnetic energy. Yet, this seemed to be impossible; the question was how could something that does not even let light escape allow heat, light or any other radiation to escape? A few interesting theories arose from this paradox. The two main theories of this phenomenon deal with particle anti-particle pairs. But first, one must realize what implications these theories cause. A problem with Black holes was that they apparently seemed to defy the second law of thermodynamics, which states that entropy, or disorder, is always on the increase. It was then realized that perhaps the event horizon was the measure of a black hole's entropy. Again a paradox arose. It turned out that as a black hole ages, it begins to evaporate, or in other words, shrink. This seemed again to defy entropy because the black hole was becoming smaller as it aged. The two hypotheses of Hawking radiation explain this.
The first, formulated by Stephen Hawking, hence the name Hawking radiation, explains that all around the event horizon of a black hole exist particle anti-particle pairs. Hawking explained that the anti-particle from the pair falls into the black hole, while the particle is emitted as radiation. What this theory explains is that as the anti-particle enters the black hole, it adds negative energy to it, thus decreasing its mass. However, the radiation emitted in the form of a particle compensates for the entropy lost while the event horizon is getting smaller.
The second hypothesis deals with virtual particles. As they "pop" up here and there next to the black hole, the black hole transfers its gravitational potential energy towards them. This process makes the virtual particles real, and so the particle anti-particle pairs annihilate themselves and give off one photon of light. This, to us, appears to be the radiation of a black hole. As the black hole gives away its gravitational potential energy to the virtual particles, it loses mass and begins to evaporate as explained. The entropy is again on the rise because there is more disorder created as the particles and anti-particles annihilate themselves.