Alpha radiation tends to happen in nuclei that have a lot of protons and a lot of neutrons. In it, the nucleus emits two protons and two neutrons as a single small nucleus, thereby changing itself into a different element. So, for example, U-238, an isotope of uranium, has 92 protons and 146 neutrons in its nucleus. After emitting an alpha particle (2 protons, and 2 neutrons), the new nucleus will have 90 protons and 144 neutrons, making it a nucleus of thorium-234.
Similarly, a Radium-226 nucleus will emit an alpha particle, starting with 88 protons and 138 neutrons and ending with 86 protons and 136 neutrons, which is a nucleus of Radon-222.
Like I said, alpha decay generally happens when there are both too many protons and too many neutrons for a nucleus to be stable. By eliminating two of each, it becomes *more* stable, but may still be radioactive.
Beta decay is a bit stranger. It happens when the number of neutrons is more than is required for stability, but the number of protons is about right. In beta decay, one of the neutrons turns into a proton and an electron is emitted (the beta particle). So, the number of neutrons goes down by one and the number of protons goes up by one.
An example is the decay of thorium-234 above. It starts with 90m protons and 144 neutrons and ends up with 91 protons and 143 neutrons. This is a nucleus of Protactinium-234.
Another example of this type of decay is that of carbon-14. It starts with 6 protons and 8 neutrons. After a beta decay, it will have 7 protons and 7 neutrons, leaving a Nitrogen-14 nucleus.
A third type of decay is called 'electron capture' and is, essentially, the reverse of beta decay: a proton 'captures' an electron surrounding the nucleus and changes into a neutron. For example, Potassium-40 can capture an electron and become Argon-40 OR it can go through beta decay and end with 20 protons and 20 neutrons, giving a calcium-40 nucleus.
One key thing here is that all of these happen in the nucleus, which is very small compared to the whole atom AND is surrounded by as many electrons as there were protons to start with. This means that ordinary chemistry doesn't have an effect on when these decays happen. Putting the atom under pressure doesn't affect the decay. Being hot or cold doesn't affect it (unless it is so hot that ALL of the electrons are pulled away, which can affect electron capture---but the temperatures required are hotter than the surface of the sun).
This is important: each type of isotope has its own rate of radioactive decay that depends on how many protons and neutrons are in the nucleus. The chemical environment, the temperature, the pressure, etc DO NOT AFFECT the rate of decay AT ALL. This has been verified with many different isotopes under a wide range of conditions.
Another important point: the atom before decay is chemically different than the one after decay. This is important because it is the chemical properties that determine what sorts of crystals the atom will join in and thereby what sorts of rocks it will be seen in. We can use this chemical information to know which atoms were NOT there when the crystal solidified: they would have been excluded by the chemical properties of the crystal.
Are we good so far?