Co-Sign vs. Co-Own vs. Convenience Signer
Typically, a co-signer is a term that you use in association with credit cards or loans, not bank accounts. It's more common to be a co-owner or convenience signer on someone's bank account. Though specific rules vary by bank, a co-signer is someone who accepts liability for the mistakes that the primary account holder makes; a co-owner has more direct control over the account and may actively manage funds; and a convenience signer has the ability to manage funds in the account, including writing checks, but does not have liability. The roles often depend on the primary account holder's age -- parents of young children will be co-owners of the account; parents with children in college will be co-signers; and children become convenience signers of their parents' accounts later in life.
Minors vs. Non-Minors
In many cases, banks will require a co-signer or co-owner for a minor to open a bank account. Parents must do this if they want their child to have an account. Those over 18, however, can usually open an account without a co-signer. If the bank requires a co-signer, it's likely because the individual has bad credit. You should be wary of co-signing an account for an adult.
The co-signer is responsible if the primary account owner defaults on the account. If the account is a savings account, it is not possible for the individual to take out more than what's in the account. However, a person can overdraft on a checking account. The bank will initially try to get the money back from the primary account holder, but if it cannot, it will turn to the co-signer, who is then responsible.
Access to Funds
The co-signer may or may not have access to the funds in the bank account. She may even be able to withdraw money without the consent of the primary account holder. Look for details regarding the rights of the co-signer in the small print of the papers you signed when opening the account.