In the days of primarily analog production, a time base corrector took an incoming video signal, stripped out the sync signals that determine the timing of the various aspects of the signal, and laid fresh sync signals with the existing video and audio.
It corrected for "drift" between stuff like the timing of the color (chrominance) part of the signal relative to the black and white (luminance) part.
They were commonly used in linear A/B roll editing systems and early NLEs that worked with analog signals.
With digital video, the timebase is typically locked relative to the chroma and luma components of the digital encoding, so TBCs are unnecessary if you're working with purely digital signal sources.
If you regularly extract old content from analog tapes - then a TBC is a great tool to put in the signal chain just before your signal gets digitized ensuring you capture the best signal possible coming off the old analog source.