What is the minimum size a black hole could be? I have been told that they were worried that the LHC would create a black hole, yet they say the Sun cannot be a black hole. I understand that the black hole may evaporate very quickly, but I still don't understand what the minimum size is.
There's no minimum size. There's a minimum density. Stars turn themselves into black holes when they exhaust their fuel and collapse. For that to yield a black hole, they need to start off with around 25 solar masses. If a star starts with enough mass, natural processes cause it to eventually suffer a core collapse which greatly compresses some of its mass. (The rest gets blasted outward into space by the resulting shockwave.) If that explosion's large enough, the mass that's falls inward can reach the critical density for a black hole. (Thus the black hole's mass is less than that of the progenitor star.)
Particle accelerators can (perhaps, according to some versions of theory) create microscopic black holes by manufacturing large subatomic particles. Since the particles are created with high nominal densities, they can be considered black holes. They also tend to be moving near the speed of light, and even though they're "large subatomic particles", they're still expected to have masses much less than, say, a Uranium atom. Since it's the total mass which sets a black hole's radius, this would yield (at worst) nucleus-scale black holes moving near the speed of light. Such black holes would drill through matter, disrupting anything encountered along the way in a nucleus-wide path. Since atoms are really small (and nuclei are even smaller), you could have a few of these going through you and not notice... but they'd technically be black holes. The "disruption" caused by them would basically be a narrow beam of ionizing radiation. Such black holes would also decay very rapidly.