Positron emission is a mode of radioactive decay very similar to beta decay, with one important difference: instead of normal electrons, positrons (anti-electrons) get emitted. In positron emission, a proton emits a positron and a neutrino and in so doing turns into a neutron:
p -> n + e+ + ν
The net result is that the atomic number of the nucleus decreases by one while the mass number remains constant, so the overall neutron/proton ratio changes in favor of neutrons.
While a free neutron will experience beta decay into a proton, positron emission can only occur for protons that are part of an unstable nucleus. The neutron is heavier than the proton, so extra energy due to nuclear instability would have to be absorbed by a proton in order for positron emission to occur.
The W boson carrying the weak nuclear force is also at work in positron emission, just as in normal beta decays. At the quark level, what happens here is that an up quark inside the proton receives some extra energy due to the general instability of the nucleus it is part of, causing it to emit a W + intermediate vector boson, which will almost instantly decay into a neutrino and a positron, while the up quark transforms itself into a down quark.
There is a related decay mode, known as electron capture, that produces the same effect as positron emission, and is more likely in elements of high atomic number. Since almost all naturally occuring radioactive elements are of high atomic number, positron emission was not observed by the early pioneers of the science of radiochemistry at the turn of the nineteenth century. The heavier elements found in nature that exhibited radioactivity preferred to change their neutron/proton ratio by capturing inner shell electrons.
Positron emission was first observed in radioactive decay by Irene Curie and Frederic Joliot in 1934, only two years after Carl Anderson confirmed the existence of the positron. They used alpha particles emitted by polonium to bombard lighter elements like boron and aluminum. Bombarding aluminum-27 with alpha particles produced a radioactive isotope of phosphorous, phosphorous-30, which underwent what they later realized was positron emission, turning into silicon-30.
Today, the phenomenon of positron emission is of great use in nuclear medicine, as it is the basis for an imaging method known as positron emission tomography. A sugar, most commonly 2-deoxy-2-fluoro-D-glucose, or 18FDG, with a positron emitting flourine-18 atom in it, is introduced into the body. The way the body metabolizes this sugar can be studied by following where this "tagged" sugar goes. See its own node for more details.
Positron emission is considerably more dangerous to humans than ordinary beta decay. A positron is an example of antimatter, so if it comes close to an electron somewhere, pair annihilation happens and two high-energy gamma rays will be emitted as a result. A radioactive isotope that is a positron emitter should thus be treated as though it were emitting hard gamma rays.
Sources:
Arthur Beiser, Concepts of Modern Physics
http://chemcases.com/nuclear/nc-02.htm
http://www.ehs.ucsf.edu/Manuals/RSTM/RSTM%20Chap1.htm
http://www.bnl.gov/pet/FDG.htm | 
