You know how before quantum mechanics people used to think that science may allow them to describe the world perfectly - that is, if they could measure all the properties of all the particles in the universe, then they could calculate its exact state both in the past and in the future.
Granted, in its pure form it was more of a popular belief than a scientific one, as any scientist worth his money should be able to recognise the actual impossibility of performing those calculations for anything but the simplest of systems(e.g.easy for two bodies bound gravitationally, insanely hard for three bodies).
Still, in principle it is true: the information a system has at any given moment encodes its past and future interactions.
I'm told that it works even for quantum mechanics, in some way.
Now, if you drop an object into a black hole, of all the information it carried with itself, like mass, chemical and elemental composition, temperature, etc., only mass, electrical charge and angular momentum are retained by the black hole(which is nicely called a "no-hair theorem"). All the rest is irrevocably lost.
So if you measure the properties of a black hole, you are unable to tell e.g.what exactly fell into it, or what kind of star's collapse produced it. All you can say is that it must've had such and such mass, charge and angular momentum.
Note that this is a paradox not in the sense that it produces logically conflicting results, but in the sense of contradicting a widely held belief about something. Here, that the information should be retained.
As for the Hawking Radiation: You know that no radiation can escape black hole, right? There's this boundary called event horizon, which marks the distance from the centre of the black hole under which the escape velocity is higher than the speed of light.
So whatever falls into that boundary should stay there forever, meaning also that black holes should only increase their masses, having no means to lose it.
Hawking came up with a way for black holes to radiate in spite of this. The idea is based on another idea, that the vacuum is full of particle-antiparticle pairs constantly appearing out of nowhere and annihilating an instant later. They pop up, fly a minuscle amount of distance and bump into each other dissapearing again.
Now, imagine these two particles appearing in such a way, that one of them flies just above the event horizon, while the other one just below. The one below is sucked in and cannot escape, while the one above now has no pair to annihilate with, AND is able to fly away from the black hole due to being above the event horizon.
Since the two particles appeared out of nothing, their total energy must be 0. Since the one flying away definitely carries some energy with it(mass, kinetic energy), then the one falling in must have negative energy. Thus it falling in causes the black hole to lose mass, which looks from the outside as if the black hole were radiating like a black body of certain temperature.