The co-signer is covered in full in the entry Cosigner, which explains the legal obligations, the effect on credit records and the risks involved. This page gives a short definition and explains the two spellings.
A co-signer is a person who signs a loan, lease or credit agreement together with the main borrower and agrees to repay the debt if the borrower fails to. Lenders ask for one when the applicant has a thin or weak credit history or low income, such as a student or a first-time renter. The co-signer usually receives no share of the money or property but is fully liable for the debt, and late payments can appear on their own credit record. The role is similar to that of a guarantor, although a guarantor's liability may arise only after the lender has tried to collect from the borrower, while a co-signer can often be pursued straight away. A comaker, by contrast, is a joint borrower who shares in the loan itself.
"Co-signer" and "cosigner" are simply two spellings of the same word; American usage increasingly drops the hyphen. Both describe the same person and the same joint liability.