Countersigning means writing a second signature

Signature

A signature is a handwritten depiction of someone's name, nickname, or even a simple "X" that a person writes on documents as a proof of identity and intent. The writer of a signature is a signatory. Similar to a handwritten signature, a signature work describes the work as readily identifying...

 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

Agency (law)

The law of agency is an area of commercial law dealing with a contractual or quasi-contractual, or non-contractual set of relationships when a person, called the agent, is authorized to act on behalf of another to create a legal relationship with a third party...

. Also, a money order

Money order

A money order is a payment order for a pre-specified amount of money. Because it is required that the funds be prepaid for the amount shown on it, it is a more trusted method of payment than a cheque.-History of money orders:...

 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

Passport

A passport is a document, issued by a national government, which certifies, for the purpose of international travel, the identity and nationality of its holder. The elements of identity are name, date of birth, sex, and place of birth....

 applications.

In some constitutional monarchies

Constitutional monarchy

Constitutional monarchy is a form of government in which a monarch acts as head of state within the parameters of a constitution, whether it be a written, uncodified or blended constitution...

 and parliamentary republics

Parliamentary republic

A parliamentary republic or parliamentary constitutional republic is a type of republic which operates under a parliamentary system of government - meaning a system with no clear-cut separation of powers between the executive and legislative branches. There are a number of variations of...

, an order by the head of state (monarch or president) is not valid unless countersigned by the prime minister or, in case of promulgation

Promulgation

Promulgation is the act of formally proclaiming or declaring a new statutory or administrative law after its enactment. In some jurisdictions this additional step is necessary before the law can take effect....

of a parliamentary resolution, the speaker of parliament.