1.
refusal; declining
Saying no to a request, an invitation or an offer; in shop signs, お断り means "not allowed" or "we do not accept".
2.
advance notice; asking permission
Telling someone beforehand or getting their permission before doing something; most often heard in negative phrases such as 断りもなく (without asking).
The noun form of 断る, which has both meanings: to refuse, and to give notice or ask permission. Not to be confused with 理 (reason, natural law), a literary word with the same reading.