Carbon-14 dating is a way of determining the age of certain archeological artifacts of a biological origin up to about 50,000 years old. It is used in dating things such as bone, cloth, wood and plant fibers that were created in the relatively recent past by human activities. Cosmic rays enter the earth's atmosphere in large numbers every day. For example, every person is hit by about half a million cosmic rays every hour. It is not uncommon for a cosmic ray to collide with an atom in the atmosphere, creating a secondary cosmic ray in the form of an energetic neutron, and for these energetic neutrons to collide with nitrogen atoms. When the neutron collides, a nitrogen-14 (seven protons, seven neutrons) atom turns into a carbon-14 atom (six protons, eight neutrons) and a hydrogen atom (one proton, zero neutrons). The carbon-14 atoms that cosmic rays create combine with oxygen to form carbon dioxide, which plants absorb naturally and incorporate into plant fibers by photosynthesis. Animals and people eat plants and take in carbon-14 as well. The ratio of normal carbon (carbon-12) to carbon-14 in the air and in all living things at any given time is nearly constant. Maybe one in a trillion carbon atoms are carbon-14. The carbon-14 atoms are always decaying, but they are being replaced by new carbon-14 atoms at a constant rate. At this moment, your body has a certain percentage of carbon-14 atoms in it, and all living plants and animals have the same percentage. As soon as a living organism dies, it stops taking in new carbon. The ratio of carbon-12 to carbon-14 at the moment of death is the same as every other living thing, but the carbon-14 decays and is not replaced. The carbon-14 decays with a half-life of 5,700 years, while the amount of carbon-12 remains constant in the sample. By looking at the ratio of carbon-12 to carbon-14 in the sample and comparing it to the ratio in a living organism, it is possible to determine the age of a formerly living thing fairly precisely. Because the half-life of carbon-14 is 5,700 years, it is only reliable for dating objects up to about 60,000 years old.

To add to the very correct answer about how Carbon-14 is formed: The carbon-14 dating method works based on the principle that the substance will decay into a more stable element over a constant period of time (constant for that element). When seen on a graph, the pattern is very distinct and consistant from sample to sample. The dating method measures the amount of radiation given off. As time passes, the amount of Carbon-14 will decrease as per the "consistant period of time." The "half-life" refered to is how long it takes for exactly half of the sample to decay into a more stable isotope. If you start with 10 pounds then at the moment of its half life, you will have half of the original amount, or 5 pounds. After the half-life time passes again, you will have 2.5 pounts, etc etc. It should be noted that after a certain period of time, Carbon-14 dating is ineffective due to small amounts approaching a negligable amount of decay. IE: stand 10 feet from a wall and step forward repeatedly, each time only covering half of the distance between you and the wall. Now, technically you'll never reach the wall, but after a certain number of steps, the amount you travel will become negligable and unmeasureable.

In addition to the other answers, there is direct counting of Carbon 14. There is a method for counting not just the beta rays from the atoms that decay but all the carbon-14 atoms in a small sample. This is particularly useful in dating very old specimens in which only a tiny fraction of the carbon 14 is left. Out of a million carbon-14 atoms, only one, on the average, will decay every three days. This makes it quite tedious, when measuring old samples, to accumulate enough counts to distinguish the radioactivity from the cosmic-ray background. But if we can count all the carbon-14 atoms now, without waiting for them to decay, we can gain a millionfold in sensitivity. This is accomplished by bending a beam of positively charged carbon atoms in a magnetic field to separate the carbon 14 from the carbon 12. The lighter carbon 12 is forced into a tighter circle, and the heavier carbon 14 is admitted through a slit into a counter. This method, although more complicated and more expensive than the beta-ray-counting method, has the advantage that the amount of material needed for a test is a thousand times less. It opens up the possibility of dating rare ancient manuscripts and other artifacts from which a sample of several grams that would be destroyed in testing just cannot be had. Now such articles can be dated with just milligrams of sample. Source: Awake! 9/22 1986

There is a nice summary of the Radiocarbon dating technique in Owen Davis' Palynology (the study of pollen) class notes at U of Arizona: http://www.geo.arizona.edu/palynology/geos462/10radiometric.html. One issue that has not been mentioned is that the amount of 14C in the atmosphere (thus absorbed by organisms) is variable. This variability includes bomb carbon (since ~1950, explaining why "14C years before present" refers to years before 1950), but also includes 14C variability driven by variations in the Earth's magnetic field, thus is screening of cosmic radiation, which forms 14C constantly from 14N. The net result is that most radiocarbon ages in the range from 12,000 to 45,000 yr B.P. are slightly too young. This doesn't mean the ages are "wrong", it means they they must be calibrated to yield the same age as K-Ar, U-series, or calendric (tree-rings, varves) ages.