What happens during a monopoly?

Depends.

In a town of 100, the local store owner is a "monopoly", but no one minds, as it's obvious that the town can't support two stores.

Then there's the Post Office, but that's the government, so monopoly laws are waived.

Then there's various religious organizations. The Catholic Church has a monopoly on who gets to be a Catholic. Of course, every other religious organization has a monopoly on who gets to call themselves a member, so no one minds. (On occassion, some do mind, which is why you see some churches called "reformed" or "restored".)

Then there's corporations that dominate their field, such that there is no meaningful competition. However, it is sometimes the case that they are just so darned good at it that no one complains, because there actually isn't anyone who could do it better or cheaper anyway. Eventually someone always complains, though.

Then, when someone complains, oft times because the originally "good" monopoly company got a bit out of hand, then the government will file an "anti-trust" suit against them. It can be settled with a voluntary correction, or in extreme cases a mandated break up can occur.