As others have pointed out, "radiation" is the energy "radiated" away from something. It could be electromagnetic radiation (radiowaves, microwaves, visible light, X-rays, gamma rays, etc.), or it could be particles (alpha or beta particles).

When we talk about washing off radioactive contamination we mean removing the radioactive particles that are emitting the radiation. If you get covered in Uranium dust, you are being exposed to all of the radiation being released by that dust. Washing it off of you removes that source of radiation thus protecting you from further exposure.

As for how there is a half life to the contamination, remember that what we're concerned with is the radioactive material itself. Radioactive material is just material that is unstable at an atomic level. That instability means it will randomly break apart, emitting energy (radiation) in the process. Once a given atom has broken apart though, it can't emit more energy.

With a large collection of radioactive material different atoms will break apart at different times, releasing their energy in a somewhat constant stream. But over time, as more of the individual atoms break apart, there will be fewer unstable atoms left. Eventually, all of the atoms will have broken apart, and so the substance will no longer be radioactive. For many things, like Uranium, this takes a very very long time though.