Hawking radiation is a process in which a black hole emits photons, and other, mostly light, particles. In this process the black hole loses its energy ie mass, and so can in principle be completely evaporated. The intensity of the radiation is inversely proportional to the mass of black holes and massive black holes radiate very poorly. For example, the black hole of mass equal to the mass of the Sun to radiate a temperature of about 107 K (Kelvin), and to completely evaporated needs 1066 years which is much longer than the age of the universe.
For this reason, radiation of black holes is not a phenomenon whose experimental observation we can expect soon. Astronomers are trying to detect radiation of super-small, so called primordial black holes which have a much higher temperature, but all efforts have remained fruitless. On the other hand, her experimental non-accessibility would compensate, the theoretical significance of this phenomenon is huge. In fact, classically speaking, neither radiation nor anyone else can escape a black hole. However, things change when we consider the quantum mechanical phenomena.
Near the black hole (but outside) from the vacuum with so called quantum fluctuations create a pair of photons or electron-positron pair, or any other particle-antiparticle pair. This violates the law of conservation of energy, but so called Heisenberg uncertainty relations allow short-term violation of that law. We can imagine that one of the two particles has a positive and another negative energy. Thereafter, the particle with negative energy falls into a black hole and "annul" the energy of black hole, and those with a positive energy leaves the area. Energy is conserved, and the ultimate outcome is that black hole radiate normal particle of positive energy that distant observer can detect. From the standpoint of the observer, the black hole is not completely black.
Excitement of this phenomenon, which is by its discoverer called Hawking radiation, lies in the fact that keys for its formation is quantum effects (vacuum fluctuations) and the gravitational effects (black hole horizon), and merger of gravity and quantum mechanics in the so-called Theory of everything is a prominent goal of modern physics. Therefore, every new theory that aspires to be a theory of everything (example, string theory) first grapple with the phenomenon of Hawking radiation, and trying to explain it. In other words, the Hawking radiation for quantum gravity might be what the black body radiation was to quantum physics.