Physically, what does it mean when people or objects are contaminated with radiation? Is it because they actually carrying heavy metal particles?
There are two distinct ways for previously non-radioactive material to become radioactive.
Contamination refers to any behavior where existing radioactive material sticks to or is incorporated in an previously non-radioactive object or body (this can include a tract of land). This requires you to go to where there is mobile radioactive material or for that material to come to you.
This is the more common way.
Activation refers to materials in a body getting transmuted by exposure to radiation. Most kinds of ionizing radiation can cause activation under the right circumstance, but it is usually a very slow process. Rare outside of labs and nuclear power facilities.
In the case of a nuclear power facility accident the scary issue is almost entirely contamination where the containment is lost material from the core (and nearby activated structural material, but mostly the core) which get carried into the wider environment by the action of water and steam. Chernobyl was special because the antique design of the reactor allow a significant conventional explosion to launch portions of the core out of the building directly.
When a person or an object is contaminated, there is some radioactive material in it (not necessarily heavy metal), either sticking on the surface or ingested.
In the context of radiation protection, you have a radioactive contamination when an unsealed radioactive source produces an unintended and abnormal level of radioactivity in the environment. Unsealed radioactive sources do not have a secure containment and are usually in the form of dissolved ions in solution (can be in gaseous form when evaporated).
Contamination most likely happens in nuclear medicine facilities where radioactive solution are used for diagnostic imaging and therapeutic purposes. An accidental spill of these solution can cause contamination in the laboratory premises.
Higher levels of contamination occur in accidents in nuclear reactors such as those used in nuclear power plants and in the production of useful radioactive materials.