A guarantor is a person who helps someone who's borrowing money by promising to pay it back to the lender if the borrower is unable to. Sometimes you also need a guarantor to rent an apartment.
If you're young, don't have a regular job, have a bad credit score, or don't have much money in the bank, you may need a guarantor in order to get a loan from a bank. The job of the guarantor is to guarantee that the loan payments will be made — by them, if necessary. You can also use this word for people who make other guarantees: "The health department is a guarantor of food safety in our city's restaurants."