Let me first give you a definition on what radioactivity is:
The spontaneous emission of radiation from the nucleus of an unstable atom.

As a result of this emission, the radioactive atom is converted, or decays, into an atom of a different element that might or might not be radioactive. I got this definition from here.

There are about 100 elements (look at a periodic table ) found here on Earth. The elements differ in their number of protons. The number of electrons is the same as the number of protons. If an element has different numbers of neutrons these variations are called isotopes (the number of protons stays the same of course, otherwise it would make it a different element). Some of the isotopes are not stable (mostly heavier elements starting with polonium). They lose energy by emitting particles and electromagnetic radiation and become a different, more stable element. The particles and energy emitted in the process are called radiation, and those unstable elements (or, isotopes) are called radioactive elements (isotopes).

There are three common types of radiation:
1. Alpha particles - Emission of Helium nuclei (2protons and 2neutrons).
2. Beta particles - Emission of electrons
3. Gamma radiation - Emission of very energetic electro-magnetic radiation.

Good examples can be found at: this site