Radiation is a general term used to relate to the way in which a tries to eliminate the energy it has. The terms ionizing and non-ionizing refer to the ability of the radiation to create ions. Ionizing radiation includes alpha radiation, beta radiation and gamma radiation. Non-ionizing radiation is electromagnetic radiation which does not have an extremely high frequency.
Emission of helium nucleus is referred to as alpha radiation. This form of radiation can cause a lot of damage due to the ions it creates.
Emission of electrons is called beta radiation. The ability to create ions is less for beta radiation than alpha radiation.
Gamma radiation is the emission of high energy photons. This is the least ionizing of alpha, beta and gamma radiation.
Radiation is the term given to an energetic particle or wave that travels through space. By energetic, we typically mean very high energy. This would be "ionizing" radiation, which means that the wave or particle has enough energy to ionize an atom. Less commonly, the term can refer to any electromagnetic wave that "radiates" from a source. The different types of radiation are classified acording to whether it is a wave or a particle, and by it's energy.
Alpha radiation is the result of decay of a large nucleus. It is a particle that is identical to a helium atom stripped of its two electrons ( so it has +2 charge). Because of its large mass, it travels relatively slowly, and is blocked easily by interactions with other atoms.
Beta radiation has two forms, + or -. Beta radiation is an energetic electron ( or positron ). It typically takes a few centimeters of a metal to stop beta - radiation (B+ radiation is antimatter, and so it annihilates on contact with ordinary matter).
Gamma radiation is electromagnetic wave with a frequency greater than 10^19 Hz. It has high energy because it is massless and carries no charge. As such, it is very difficult to block gamma radiaiton.
X-Ray radiation is electromagnetic wave with a frequency from about 10^16 to 10^19 Hz. The wavelength of X-Rays is comparable to the size of larger atoms. Hence, larger atoms (like calcium) tend to absorb X-Rays, while smaller atoms, like carbon, tend to let the x-rays pass. This behaviour permits the x-ray images we see in doctor's offices.