A blank check (blank cheque, carte blanche), in the literal sense, is a check that has no numerical value written in, but is already signed. In the figurative or metaphoric sense, it is used (especially in politics) to describe a situation in which an agreement has been made that is open-ended or vague, and therefore subject to abuse, or in which a party is willing to consider any expense in the pursuance of their goals.
Literal meaning 
Check users are normally advised to specify the amount of the check before signing it. If created accidentally, a blank check can be extremely dangerous for its owner, because whoever obtains the check could write in any amount of money, and would be able to cash it (to the extent that the checking account contains such funds, also depending on the laws in the specific country).
One might give a blank check to a trusted agent for the payment of a debt where the writer of the check does not know the amount required, and it is not convenient or possible for the writer to enter the amount when it becomes known. In many cases, it is possible to annotate a check with a notional limit with a statement such as "amount not to exceed $1000". In theory, the bank should refuse to process a check in excess of the stated amount.
The "formal" American legal term for a blank check is an incomplete instrument – rather, a blank check is an example of an incomplete instrument, which more generally is any incomplete signed writing – and these are covered in the Uniform Commercial Code's Article 3, Section 115. Filling in an amount into a blank check, without the authority of the signer, is an alteration (covered in Article 3, Section 407), and is legally equivalent to changing the numbers on a completed (non-blank) check, namely that the check writer is not liable for the check. However, the check writer has the burden of proving that the alteration was not authorized.
Counter check 
Blank check was also commonly used as a synonym for counter check. Before the Federal Reserve established regulations in 1967 requiring that check be MICR encoded in order to be handled by their clearing houses, it was fairly common for banks, especially in small towns, to issue check to customers which were not personalized other than the name of the bank.
Businesses would have pads of counter check which did not even have the bank specified on them - the customer had to not only fill in the value of the check, the date, and their signature, but also had to designate the bank on which funds were to be drawn.
Metaphoric meaning 
The metaphor of the "blank check" is thus often used in politics. For example, in the United States, the Gulf of Tonkin Resolution has been called a blank check as it gave the President, Lyndon B. Johnson, the power to "take all necessary measures" to prevent "aggression" in Southeast Asia. These powers were then used to escalate the Vietnam War. Many in the United States Congress protested, but were helpless to effect change, for the Tonkin resolution's terms were too subjective to enforce.
This term was also used to describe how the Kaiser of Germany (Kaiser Wilhelm II) told Austria-Hungary officials that they could deal with Serbia however they wanted after Serbian Nationalists assassinated the heir to the throne of Austria-Hungary, Archduke Franz Ferdinand. This immediately preceded World War I.
An example of the second metaphorical usage can be seen in a BBC News article, in which Gordon Brown, the then Chancellor of the Exchequer, offered a 'blank check', and would thus '"spend what it takes" to tackle Iraq's weapons of mass destruction.'
It may also be used in service fields. Customers may tell a company to treat the project as their own, which, in essence, is a carte blanche. (To the extent the service meets normal expectations.)
After the September 11 attacks, many described President George W. Bush's reaction as giving the federal government a "blank check" in order to hunt down those responsible to the attacks and prevent such attacks from happening again.
In literature 
Dec. 3, 1627 It is by my order and for the good of the state that the bearer of this has done what he has done. Richelieu
or in French:
3 décembre 1627. C'est par mon ordre et pour le bien de l'Etat que le porteur du présent a fait ce qu'il a fait. Richelieu.
"Your Majesty will, of course, stay in London for the present?” “Certainly. You will find me at the Langham under the name of the Count Von Kramm.” “Then I shall drop you a line to let you know how we progress.” “Pray do so. I shall be all anxiety.” “Then, as to money?” “You have carte blanche.” “Absolutely?” “I tell you that I would give one of the provinces of my kingdom to have that photograph.”
"I understand that you give me carte blanche to act for you, provided only that I get back the gems, and that you place no limit on the sum I may draw.” “I would give my fortune to have them back.”
Blank check company 
In economics, the term blank check company can refer to a company in development that has no specific business plan yet. For a fuller discussion of blank check companies, see Special purpose acquisition company.