Carbon occurs in different isotopes, including 12C, 13C and 14C. The first two are non-radioactive, while 14C is radioactive and decays into 14N, with a half life of 5730 years. Because of the short half life, any 14C present at the beginning of the Earth has long since disappeared, and it would not even exist today, if there were not a regenerating mechanism that constantly produces a small amount due to the cosmic ray bombardment of the upper atmosphere. Even so, the amounts of 14C are so small that we can ignore it for most purposes, unless we are looking specifically for it (e.g., radiocarbon dating).
Of the other two isotopes, about 99% of the carbon in the universe (and Earth) is 12C, while the remaining 1% is 13C. Although isotopes are considered to be identical so far as their chemistry goes, it so happens that the enzymes involved in photosynthesis have a slightly greater affinity for 12C than 13C. For this reason, any biogenic carbon (carbon that is part of a living organism or its remains) is slightly lighter than carbon that is not biogenic. The remaining, non-biogenic carbon, by contrast is isotopically heavy, because more of the lighter 12C has been extracted by living organisms from it than the heavier 13C.
Remember that as biological material does not stay put, it is constantly recycled through the carbon cycle. So when an organism dies, unless its remains are sequestered in some way (as in a fossil) and prevented from mixing with its surroundings, biogenic carbon will disperse into the environment and gradually assume the same isotope ratios as inorganic carbon.
Mass spectrometry can easily determine very small variations in the 12C : 13C ratio, and these ratios are commonly used on inorganic carbon. A positive 13C excursion happens when a lot of 12C is locked up in biological material, and as a result the rest of the carbon (such as carbon dioxide in the atmosphere) becomes heavy with 13C. There is often great debate regarding the reasons for such excursions, but the fact that the excursion happened can be easily measured with a high degree of repeatability.
Figure showing 13C excursions during the Proterozoic and Phanerozoic. The arrows indicate excursions that occurred during the two major glaciations of the Cryogenian period of the Neoproterozoic - the Sturtian and Marinoan glaciations. Adapted from Bartley and Kah, 2004.
The figure shows the carbon ratios during the Proterozoic and Phanerozoic eons. The arrows indicate excursions that took place during the Cryogenian period, and are associated with two major glaciations that have been implicated in the "snowball Earth" scenario.
The cause for these excursions is a matter of intense debate. The carbon cycle is complex even today, and how it worked in the distant past is a matter of conjecture. Many attempts have been made to relate it to geological events such as the snowball Earth (when in many ways, transfer of material between different parts of the Earth, such as between continents and oceans) was greatly impeded, or volcanism accompanying the end of the glaciations, when large amounts of carbon dioxide was released into the atmosphere. However, the data doesn't provide a perfect fit to any of these explanations, and investigations continue.
Note that the end of the Neoproterozoic and the early Phanerozoic are a period of great instability in this history, with carbon ratios constantly rising and falling. This record is difficult to interpret.