Yes, it is believed that black holes can evaporate.
The 'evaporation' of black holes follows from a theory called Hawking Radiation, which, of course, was put forward by Stephen Hawking. This result largely draws from quantum physics. The most common explanation of Hawking Radiation is as follows:
Virtual particle pairs of matter and anti-matter keep getting created in vacuum all the time. Generally, they annihilate each other quickly and vacuum remains vacuum. However, when this happens near the event horizon of the black hole, and one particle falls into it (visualize the black hole 'sucking' one particle), while the other stays outside (escapes the black hole) without getting annihilated, it appears as though a new particle (the escaped particle) has suddenly appeared out of 'nowhere'! This is a flagrant violation of the law of conservation of mass and energy. So, in order to preserve the law (the laws of physics are never broken, at least in our universe), the particle that fell into the black hole is said to have a negative mass. Now, a particle with a negative mass fell into a black hole. So, the mass of the black hole must have decreased.
The particle that escaped the black hole might as well have just been 'emitted' by the black hole, thus causing the mass-energy of the black hole to decrease.
If black holes keep radiating particles this way, they could, theoretically speaking, disappear some day (a day far, far, far away!).More questions on Black Holes: