A surety is one who, at the request of another, and for the purpose or securing to him a benefit, becomes responsible for the performance by the latter of some act in favor of a third person
e.g., A promises B that A will pay C's debt. C only has an obligation to B.
Promising the obligor that you'll pay off his debt is not a suretyship.
Co-signing a loan makes you a co-obligor, not a surety.