Radioactivity occurs simply when very small particles in objects emit energy or smaller particles. The energy that is produced can cause cancer, serious environmental damage, or useful technologies. There are different degrees of radioactivity and different exposures increase the damage it can cause. Radioactivity is a characteristic of certain types of matter.
All matter is made of chemical elements and elements are made of atoms. I mean, they don't change over time. However, radioactive atoms change over time. Small particles and energy fly out of them naturally.
The particles and energy that are released are a form of radiation. If an element or isotopes of that element are said to be radioactive, it means that they emit some kind of radiation.