Radiation is a catch-all word used to describe every form of energetic particle or electromagnetic wave found in the universe.
It is also used in thermodynamics to describe one of the ways by which energy is transferred, along with conduction and convection. This is referred to by physicists as "black body radiation," and by lay people as "radiators."
Colloquially radiation is "bad," implying death, disease, and destruction. Where once it was hailed as the "cure all" it is now seen as the "killer."
The most common form of radiation is photons. This includes everything from radio frequency energy, which can be emitted even at the low temperatures found in interstellar dust clouds, through the heat of any warm body radiating into the surroundings, on through thermal infrared, light, X-rays and gamma rays emitted by stars and other energetic bodies.
The effect on matter of these radiations is dependent on the energy of the individual photons, which is directly proportional to the frequency. Frequencies lower than X-rays are normally regarded as non-ionizing and not affecting matter, but fortunately this is not the case - without the natural effects of this radiation, life (as we know it) would not exist. "Visible light" frequencies are used by plants to accomplish photosynthesis. It is the reaction of light only slightly more energetic than visible (the ultraviolet) which can cause melanoma by reacting with epidermal cells. Radio frequency radiation is used in radio, television, cellular phones, radar, microwave ovens, etc. All these applications rely on the reaction of some atoms, molecules, or crystals to the particular frequency involved. All forms of electromagnetic radiation, including visible light, can produce radiation burns if sustained at a high enough level.
 Ionizing radiation
Ionizing radiation is electromagnetic radiation of sufficiently high frequency -- that is, sufficiently high energy per photon -- to completely dislodge an electron from the atom or molecule it's bound to. Hard ultraviolet, X-rays, and gamma rays occupy the higher frequency end of the electromagnetic spectrum, and are ionizing radiation. Soft ultraviolet, visible light, infrared light, and radio are not.
Ionizing radiation can have a direct effect on matter by dislodging one or more electrons from its atom or molecule, leaving a reactive ion in its wake. It is the existence of these ions within cells that can cause mutation in DNA, sometimes leading to cancer or organ dysfunction.
Alpha particles consist of two protons and two neutrons, and so are indistinguishable from nuclei of helium atoms (they are often said to be the nuclei of helium atoms but are usually the result of radioactive decay or similar processes and have never existed in a heterosexual marriage with two electrons as a helium atom). Beta particles are high speed free electrons.
Fast free neutrons are also classified as ionizing radiation. Slow free neutrons are more problematic. Slow, or "thermal", neutrons can be captured by the nucleus of whatever atom they happen to bump into, resulting in the atom becoming a heavier isotope of the same element. A cobalt-59 atom, for example, can absorb a neutron and become cobalt-60. This is how radioactive contamination occurs. It's important to note, however, that only a neutron source can cause a non-radioactive item to become radioactive. "Nuclear" substances that don't emit neutrons, such as the cobalt-60 used in food irradiation (which only emits gamma rays and beta particles), cannot cause radioactive contamination.
 Natural sources of radiation
 Black body radiation
Photon radiation (electromagnetic) is emitted by all bodies in the universe. The hotter the body, the higher the frequency, that is: the energy of the emitted radiation is proportional to temperature. Stated another way, the wavelength is inversely proportional to the temperature. At high enough temperatures, this is manifested as visible light. The term "red hot" precisely encapsulates this concept: at a particular range of temperatures, the radiation emitted has the frequency/wavelength we associate with the color red.
Objects at "normal" temperatures, like trees and people, emit most of their blackbody radiation in the far infrared. Thus, this portion of the infrared spectrum is sometimes called the "thermal infrared." It is this portion of the spectrum that thermograph cameras pick up.
Black body radiation shows us that eventually, the universe will reach thermal equilibrium, since any object warmer than any other will transfer some net amount of energy to the cooler one by this process.
 Nuclear decay
All atomic reactions, i.e. atomic fission, or fusion, will emit energy. Even when there is a net reduction in free energy there will be some emission either of photons or of sub-atomic particles. Many such reactions will produce a multiplicity of particles, which will decay in short periods of time into photons, alpha particles, neutrons or electrons.