Radioactive decay is a change in an atom's nucleus that happens to individual nuclei by chance: with a fixed rate for a given nuclide (i.e., for a given combination of protons an neutrons, or equivalently, for a given atomic number and mass number). The tendency of some nuclei to decay in this manner is termed radioactivity.
If you had a given number of a particular radioactive nuclide, and each time one decayed, you replaced it with another (non-decayed), the long-term rate of decays (average count of decays per second) would remain constant. Without such replacement, the number of nuclides not yet decayed (parent nuclides) is constantly falling exponentially so the rate of decays falls with it. Given this fact, decay rates are often characterized by half lives, the time it takes for half the original parent nuclides to decay into the resulting nuclides (daughter nuclides), or alternately, by a radioactive decay timescale, the time in which the count of parent nuclides falls by a factor of e.
Radioactivity generates heat (radioactive heating), can damage surrounding material due to particles expelled in the process, and always changes the material's chemical composition, i.e., the resulting atomic numbers are different. The latter can be observed at a distance through spectroscopy: for example, some supernova produce radioactive elements and observable spectral lines for the parent and daughter nuclides allow the decay and its rate to be detected.
A material's composition shows signs of any radioactivity, i.e., isotopes in otherwise-unexpected ratios. By extrapolating backward, the age of materials can be determined (radioactive dating).