Doing something "indefinitely" means doing it without an end point.

For example, imagine that a police officer gets suspended from her job because she did something wrong:

A: How long is she suspended for?

B: Indefinitely.

This means that a time hasn't been set for her to return to her job. She might be allowed to return in a few weeks, or she might never be allowed to return.

The word "indefinitely" is a little technical-sounding, so people use it for talking about serious topics.