The bomb pulse is a big increase in radioactive carbon (C14) in the environment, especially from the atmosphere but also in plants and animals, from above-ground nuclear testing in the 1950s. When your body puts carbon into DNA, such as when replicating a cell, it doesn't care whether it's C14 or regular carbon, but whatever gets put there never changes. You can measure C14 using mass spectrometry, so you can indirectly measure the age of the cell. (I'm not really clear on the details of how that works.)
Here's a NOVA episode that talks a lot about the technology and uses. Warning, some of the research related to brain function is a bit gross.
Cold War Bomb Testing Is Solving Biology’s Biggest Mysteries
I'm not real clear on how it helps with identification. It sounds like maybe everybody has a different C14 "carbon fingerprint" or something.