Short answers, pulled from the story.
Radiocarbon dating was developed by Willard Libby in the late 1940s at the University of Chicago. Libby published his foundational paper in Science in December 1949 and received the Nobel Prize in Chemistry for this work in 1960.
Radiocarbon dating is generally reliable for samples up to about 50,000 years old, after which too little carbon-14 remains to detect accurately. Special preparation techniques and very long measurement times can occasionally extend the range to around 60,000 or even 75,000 years before the present.
The currently accepted half-life of carbon-14 is 5,700 plus or minus 30 years. For consistency with historical results, radiocarbon ages are still calculated using Libby's original value of 5,568 years, known as the Libby half-life; calibration curves correct for this discrepancy in final reported dates.
Beta counting detects the radioactive decay of individual carbon-14 atoms and requires a sample of at least about 10 grams. Accelerator mass spectrometry directly counts the carbon atoms present and can work with samples as small as 0.5 milligrams, achieving one percent accuracy in minutes rather than hours.
Marine organisms absorb carbon from ocean water that is a mixture of recently surfaced water and much older deep-ocean water. Because the deep ocean takes about 1,000 years to circulate back to the surface, the surface water is depleted in carbon-14, giving marine life an apparent radiocarbon age roughly 400 years greater than their true age on average.
Radiocarbon dating provided the first time scale that could link archaeological sites across continents, which historians of the field call the radiocarbon revolution. It revealed that many prehistoric innovations arose independently in different regions rather than spreading by migration, and the later development of accelerator mass spectrometry opened research on questions that were previously unanswerable with the sample sizes available.