Lead is a very dense material, and is much cheaper than Gold or Silver, and is easier to work with and less toxic in common forms than Thallium or Bismuth (which are right nearby on the periodic table). It is also more readily available. Here's why the density of Lead is so useful.
Radiation comes in three main types, Alpha, Beta, and Gamma. Alpha radiation can be stopped by a piece of paper or the surface of your skin. An Alpha particle is essentially a highly charged Helium nucleus, and it doesn't travel very far. Beta radiation is a high-speed electron, which travels further, but can be stopped with a sheet of aluminum foil. Gamma radiation, however, is a high-energy photon. Because it is a photon, and because it has so much energy, they are hard to stop. 30 mm of lead will do the trick.
Matter is not as solid as it looks when you get down to the atomic scale. The denser a particular material, the more atoms it has per unit of volume. The more actual atoms in the material, the higher the chances that the gamma ray photon will collide with one of them, thus expending it's energy on the lead, and not on your body.
Certain types of radiation also give off neutrons, and since neutrons are neutral in charge, they aren't attracted to anything in the material. Again, the density of the lead helps increase the probability of matter collisions that reduce the energy of the particle.
Frank DiBonaventuro, B.S., Air Force Officer, Physics grad, The Citadel