Alpha decay
Alpha decay is a type of radioactive decay in which an atomic nucleus emits an alpha particle (two protons and two neutrons bound together into a particle identical to a helium nucleus) and transforms (or 'decays') into an atom with a mass number 4 less and atomic number 2 less. For example:
{}^2{}^{38}_{92}\hbox{U}\;\to\;{}^2{}^{34}_{90}\hbox{Th}\;+\;{}^4_2\hbox{He}^{2+},
although this is typically written as:
{}^{238}\hbox{U}\;\to\;^{234}\hbox{Th}\;+\;\alpha.
(The second form is preferred because the first form appears electrically unbalanced. Fundamentally, the recoiling nucleus is very quickly stripped of two electrons to neutralize the ionized helium cation.)
An alpha particle is the same as a helium-4 nucleus, and both mass number and atomic number are the same.
Alpha decay is a form of nuclear fission where the parent atom splits into two daughter products. Alpha decay is fundamentally a quantum tunneling process. Unlike beta decay, alpha decay is governed by the strong nuclear force.
Alpha particles have a typical kinetic energy of 5 MeV (that is 0.13% of their total energy, i.e. 110 TJ/kg) and a speed of 15,000 km/s. This corresponds to a speed of around 0.05c. Because of their relatively large mass, +2 charge and relatively low velocity, they are very likely to interact with other atoms and lose their energy, so their forward motion is effectively stopped within a few centimeters of air.
Most of the helium produced on Earth comes from the alpha decay of underground deposits of minerals containing uranium or thorium.  The helium is brought to the surface as a by-product of natural gas production.

# History
By 1928, George Gamow had solved the theory of the alpha decay via tunneling. The alpha particle is trapped in a potential well by the nucleus. Classically, it is forbidden to escape, but according to the then newly discovered principles of Quantum mechanics, it has a tiny (but non-zero) probability of "tunneling" through the barrier and appearing on the other side to escape the nucleus.

# Uses
Americium-241, an alpha-emitter, is used in smoke detectors. The alpha particles ionize air between a small gap.  A small current is passed through that ionized air. Smoke particles from fire that enter the air gap reduce the current flow, sounding the alarm.
Alpha decay can provide a safe power source for radioisotope thermoelectric generators used for space probes and artificial heart pacemakers. Alpha decay is much more easily shielded against than other forms of radioactive decay. Plutonium-238, for example, requires only 2.5 mm of lead shielding to protect against unwanted radiation.
Static Eliminators typically use Polonium-210, an alpha emitter, to ionize air, allowing the 'static cling' to more rapdily dissipate.

# Toxicity
Being relatively heavy and positively charged, alpha particles tend to have a very short mean free path, and quickly lose kinetic energy within a short distance of their source.  This results in several MeV being deposited in a relatively small volume of material. This increases the chance of cellular damage in cases of internal contamination. In general, external alpha radiation is not harmful since alpha particles are effectively shielded by a few centimeters of air, a piece of paper, or the thin layer of dead skin cells. Even touching an alpha source is usually not harmful, though many alpha sources also are accompanied by beta-emitting radiodaughters, and alpha emission is also accompanied by gamma photon emission. If substances emitting alpha particles are ingested, inhaled, injected or introduced through the skin, then it could result in a measurable dose.
The Relative Biological Effectiveness (RBE) is a measure of the fact that alpha radiation is more effective at causing certain biological effects, notably either cancer or cell-death, compared to photon or beta radiation, for equivalent radiation exposure.  This is generally attributable to the high Linear Energy Transfer (LET), which is about one ionization of a chemical bond for every Angstrom of travel by the alpha particle.  The RBE has been set at the value of 20 for alpha radiation by various government regulations.  The RBE is set at 10 for neutron irradiation, and at 1 for beta and ionizing photon radiation.
However, another component of alpha radiation is the recoil of the parent nucleus, due to the conservation of momentum requiring the parent nucleus to recoil, much like the 'kick' of a rifle butt when a bullet goes in the opposite direction.  This gives a significant amount of energy to the recoil nucleus, which also causes ionizaton damage.  The total energy of the recoil nucleus is readily calculable, and is roughly the weight of the alpha (4 amu) divided by the weight of the parent (typically about 200 amu) times the total energy of the alpha.  By some estimates, this might account for most of the internal radiation damage, as the recoil nuclei are typically heavy metals which preferentially collect on the chromosomes.  In some studies, this has resulted in a RBE approaching 1,000 instead of the value used in governmental regulations.
The largest natural contributor to public radiation dose is radon, a naturally occurring, radioactive gas found in soil and rock.   If the gas is inhaled, some of the radon particles may attach to the inner lining of the lung.  These particles continue to decay, emitting alpha particles which can damage cells in the lung tissue.. The death of Marie Curie at age 66 from leukemia was likely caused by prolonged exposure to high doses of ionizing radiation. Curie worked extensively with Radium, which decays into Radon, along with other radioactive materials that emit beta and gamma rays.
The 2006 assassination of Russian dissident Alexander Litvinenko is thought to have been caused by poisoning with Polonium-210, an alpha emitter.