Which type of radiation can penetrate a thin piece of lead?

Yes, but anything will block all radiation, if you use enough of it. Lead, being very dense, is very efficient in blocking most radiation even in fairly modest thicknesses. For example, when you go to the dentist for an X-ray, the radiologist covers you with a lead-foil apron.

The very best material to block radiation, in terms of cost, is about a half-mile thickness of dirt. Cheap, and very effective. However, it is difficult to transport. Some types of radiation, like alpha particles, can be blocked very effectively by a sheet of paper. Electromagnetic radiation such as gamma rays or X-rays requires a material with high density to block it.