I think it means to write a bank check, in order to pay money to someone. Actually, I had to google it, because I've never heard this phrase before. I assume it's American English, rather than British English. In British English, the phrase would be "to write someone a cheque" (note the different spelling of "check" in American English and "cheque" in British English). Can any Americans confirm this? (I may be wrong!)

It's common in US and has a distinct meaning from writing a check. "Cut a check" is when a check is issued, specifically printed. We don't generally say "cut" for a handwritten check, we say "write a check". So typically "cut" is for business checks. If you don't understand the word check/cheque, it's a note that you fill out that tells your bank to give someone money. "I'll have the accounting department cut you a check in the morning"