Countersigning means writing a second signature onto a document. For example, a contract or other official document signed by the representative of a company may be countersigned by his supervisor to verify the authority of the representative. Also, a money order or other financial instrument may be signed once upon receipt, then signed again by the same person when presented for payment, as an indication that the bearer is the same person who originally received the item, and not a thief who has stolen the item before it could be carried to the place where it was to be presented.
An example of where a countersignature is needed is with UK passport applications.
In some constitutional monarchies and parliamentary republics, an order by the head of state (monarch or president) is not valid unless countersigned by the head of government or, in case of promulgation of a parliamentary resolution, a presiding officer of the legislature.