What is radioactivity?
Radioactivity is the emission of energetic particles or waves from atoms. Natural radiation occurs when unstable nuclei transform to some other nucleus by emitting radiation. Induced radiation often occurs when electrons which have been excited lose energy in the form of X-rays or microwaves, as done in dentist offices, microwave ovens, and radar. As it applies to nuclear energy, many materials created during the operation of a reactor are unstable. As they decay over varying lengths of time (from microseconds to hundreds of thousands of years), they emit energetic particles or waves.
If you’re looking for math, see the math behind radioactive decay page.
Types of nuclear radiation
Named alpha because they were the first to be discovered, these particles are made up of 2 protons and 2 neutrons: the helium nucleus. Often, large atoms decay by emitting an energetic alpha particle. These particles are relatively large and positively charged, and therefore do not penetrate through matter very well. A thin piece of paper can stop almost any alpha particle. However, the particles cause extreme damage of materials that they stop in by displacing atoms as they slow. Paper under sustained alpha-irradiation would degrade.
Beta particles are energetic electrons that are emitted from the nucleus. They are born when a neutron decays to a proton. Since neutrons are neutral particles and protons are positive, conservation of charge requires a negatively charged electron to be emitted. Some isotopes decay by converting a proton to a neutron, thus emitting a positron (an anti-electron)[*]. These particles can penetrate matter more than can alpha particles, and it takes a small aluminum plate to stop most beta particles.
Gamma rays are photons that are emitted from the nucleus. Often an atom in an excited state will de-excite by emitting a gamma ray. Gamma rays are similar to light waves and x rays, except they are usually much higher frequency and consequently, more energetic. This radiation has no charge, and can penetrate most matter easily, requiring a lead brick for shielding.
Common sources of radiation
Smoke detectors make use of the isotope Americium-241. This isotope emits alpha-particles at energies up to 5.4 MeV. The energetic alpha particles are used to ionize air. Once the air is ionized, a small current runs through it. When smoke enters the chamber, the current experiences an increase in resistance and a circuit sounds the alarm.
Radioactive isotopes (radioisotopes) are commonly used for medical care. Positron emitters can be introduced into the body, where the positron annihilation reactions are then monitored by sophisticated detections systems that can reconstruct intricately detailed images of the body in a PET[*] scan.
Coal-burning power plants
Coal is an impure fuel, and it usually contains 1.3 ppm of uranium and 3.4 ppm of thorium (not to mention arsenic, mercury, and sulfur). When coal burns, these isotopes are emitted into the atmosphere, where they enter our ecosystem. This leads to the astounding fact that the population effective dose equivalent from coal plants is 100 times that from nuclear plants.
Nuclear weapon detonations
The numerous atmospheric nuclear weapons tests that have occurred in the last decade have left long-lived radioisotopes in the atmosphere. The US alone has conducted at least 331 of these tests. 
This natural occurring gas comes from soil and is found throughout the world. It emits alpha particles, and can therefore damage DNA and lead to cancer if inhaled. The EPA recommends you check your house for radon gas.
Cosmic rays are energetic particles that originate outside of earth, in the sun, distant stars, galaxies, and supernovae. Most of these are protons. The atmosphere shields us from most cosmic rays, but during air travel, one will accumulate much higher dose 
Video of radiation detection
Watch the a video of a few of us detecting radiation from household items.
Background readings in Ann Arbor, MI
For a class in 2005, with no radioactive sources within range, we measured a long (30 minutes +) reading with a high-purity germanium (HPGe) gamma-ray detector system. We then identified the source of each peak. The spectrum is shown in the figure. Click it for the identifications. HPGe detectors are known for excellent resolutions, and as you can see, many peaks are clearly visible. Each one represents a specific nuclear reaction. Some major gamma-rays are highlighted on the figure. Thallium-208 is a decay-product of Thorium-232, which is naturally present in soil. Protactinium-234 results from the natural alpha-decay of Uranium-238. Potassium-40 is found all around, including in bananas and in salt-substitutes at the grocery store.