Radioactivity

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Because a nucleus is made of protons, there has to be a nuclear strong force to keep them together, since they would naturally repel each other. Neutrons in the nucleus also help by keeping the protons farther apart from each other and "diluting" the nucleus.
But when there are too many protons, the proton-proton repulsions build and the nucleus becomes unstable. It carries more energy than other arrangements of nucleons accessible to it. To achieve less energy (and more stability), they undergo radioactive decay, ejecting small nuclear fragments and high-energy radiation.
There are three main types of radiation, alpha, beta and gamma.

Alpha

Alpha radiation is the emission of a helium particle stripped of its electrons. The resulting nucleus is missing 2 protons and 2 neutrons resulting in a charge loss of 2 and mass loss of 4. An alpha particle has a charge of +2. The massive kinetic energy emitted by an alpha particle, upon impact with near by particles, usually causes reactions in those molecules. The largest radiation particle, alpha particles are capable of doing the most damage. However because of its size it cannot penetrate our skin. When the an alpha emitter is in gaseous form it becomes threatening, it no longer has to penetrate skin, it can proceed directly into the lungs causing cancer of the lungs.
For example, this is the alpha decay of uranium-238 to thorium-234: 
23892U ==> 23490Th + 42He
(When doing nuclear equations, make sure that the mass numbers (the top) and the atomic numbers (the bottom) on each side are equal.)

Beta

Beta radiation is the emission of high energy electrons from the nucleus of an atom. The loss of an electron does nothing to the atomic mass but increases the atomic number by 1. Basically, beta radiation changes a neutron into a proton. The charge of a beta particle is -1. Beta radiation is more penetrating then alpha radiation but lacks the kinetic energy of alpha particles. The wavelength of beta rays is approximately that of X-rays, 10-8 m.
This is the general equation for beta decay: 
10n ==> 11p + 0-1e + antineutrino
neutron ==> proton + beta particle (electron) + antineutrino
An antineutrino accompanies the products also.

Gamma

Gamma radiation (00) is the emission of high energy photons in the range of 10 -10 m, with no matter associated with it. Gamma rays are normally the by product of other alpha or beta emissions. Gamma rays have no effect on either mass or charge, gamma rays only stabilize the nucleus by releasing some of the excess energy. Gamma rays are the most damaging rays being able to penetrate meter thick lead. High energy gamma rays are known as cosmic rays are capable of penetrating kilometers of dense material.

Other Types

A few synthetic isotopes emit positrons, the antimatter equivalent of an electron (beta particle). A positron is an electron with a positive charge. In positron emission, a proton changes into a neutron, in this general equation: 
11p ==> 10n + 01e + neutrino
proton ==> neutron + positron + neutrino
Notice that the products of this, the positron and the neutrino, are antimatter equivalents to the electron and antineutrino in the beta emission.
When matter and antimatter come together, their masses change into the energy of two gamma-ray photons called annihilation radiation photons. 
0-1e + 01e ==> 200
Neutron emission leads to a different isotope of the same element, for example: 
8736Kr ==> 8636Kr + 10n
In electron orbital capture, a very rare type among natural isotopes but common among synthetic ones, a nucleus can capture orbital K shell or L shell electrons. In other words, it is the collapse of an orbital electron into the nucleus, changing a proton into a neutron, and emitting X rays and neutrinos. For example: 
5023V + 0-1e ==> 5022Ti + X rays + neutrino

Half-Life

When unstable nuclei undergo radioactive decay, their decay rate is not steady. Instead, they have a half-life (t1/2), which is the amount of time required for half of the reactant to disappear. For example, Iodine-131 is unstable and under beta particle emission, it becomes Xenon-131. It has a half life of roughly 8 days. So if we had a 4 gram sample of Iodine-131, in 8 days, there would be 2 grams left. In another 8 days, there would be 1 gram.
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