Countersigning means adding a signature to a document that has previously been signed by someone else. The purpose of countersigning a check is usually to cash it or to deposit it.
Two main uses of countersignatures on checks are to cash a third-party check and to cash a traveller's check.
A payee (the person to whom a check is written) may under certain circumstances use a check written to the payee to pay another person. To do this, the payee must endorse the check, writing "pay to the order of" and the second person's name on the back of the check. That second person may then endorse the check, or countersign, and deposit it. Some financial institutions will not accept third-party checks.
Countersigning a traveller's check consists of dating and signing a traveller's check in the presence of the person accepting the check. When you purchase a traveller's check, you place your signature on it. When you countersign, the person accepting the check will verify that the signatures match.
Parties should only agree to deposit third-party checks for trustworthy individuals who are known to them. Depositing a third-party check could result in the payee's (the countersigner) being liable for the amount of the check, if the check was written with insufficient funds.