Radioactivity: Spontaneous changes in a nucleus accompanied by the emission of energy from the nucleus as a radiation.
Radioactive Half-Life: A period of time in which half the nuclei of a species of radioactive substance would decay.
We imagine that we have a radioactive substance. When the nuclei of the substance decay, they emit radiation (alpha, beta, or gamma rays) that can be detected by counters such as a Geiger Counter. For a Geiger Counter, each time an emitted particle passes through the Geiger Counter, the counter makes a clicking sound. The number of clicks per unit time of the counter tells us how many decays per unit time are occurring. But the rate of clicks decreases with time because the rate is directly proportional to the number of radioactive nuclei in the substance that can decay. Hence, as time goes on, we know that the number of radioactive nuclei in the substance must also be decreasing.
The radioactive half-life of the substance is the period of time over which the number of radioactive nuclei decreases by a factor of one-half.
Radioactive decay is a quantum mechanical process governed by probability waves. In a short period of time, each radioactive nucleus has a certain probability of decaying, but whether it actually does is determined by random chance. In the animation, we plot the number of remaining undecayed nuclei as time goes on. There is some irregularity introduced into this plot by the quantum randomness. When the animation repeats itself, you will see that it does not happen the second time in exactly the way it did the first time.