Watching someone else's child while the parents are away is called "babysitting".
When you use the word "babysit" to talk about an adult, though, it's a little insulting. It means that you have to watch the adult for some reason like:
- making sure that the person doesn't make a mistake in their job
- making sure that a socially awkward friend or partner is having fun at a party
People say things like:
I don't have time to babysit you. Figure it out on your own!