A cheque is a type of document that is commonly used in Banking. Its primary purpose is to tell the bank to withdraw the amount from the drawer's account and to pay that amount to the payee. The drawer is the person who signs the cheque and from whose account the money would be withdrawn from; basically he is the one who need to pay money to somebody else. The payee is the person who gets the money; they are the ones in whose name the cheque is make.

Now, a check is the same exact thing as a cheque. In fact, a cheque is known as a check. The only difference is that it is known as a cheque in British Language, while it is known as a check in American English. The only thing that varies is the spelling of the word; the meaning, the usage, and even the pronunciation is exactly the same.

However, the term check also has a variety of different meaning which are not related to banking. According to the Oxford Dictionaries, some of these definitions are:

• Examine (something) in order to determine its accuracy, quality, or condition, or to detect the presence of something. E.g. Let me check the schedule
• Stop or slow the progress of (something, typically something undesirable) E.g. Check the disease
• Consign (baggage) to the care of the transport provider with whom they are travelling. E.g. Check the suitcase at the airport
• Mark or click on (a box) in order to select a particular option on a form, questionnaire, etc. E.g. Check the box that applies to you
• The term is also used in Chess to indicate when the king is in danger

However, there is no other definition for cheque. Even in British English, the term cheque is only used in the context of banking and finance; in all other instances, including the ones listed above, check is used.

Comparison between Cheque and Check:

Reference: Oxford Dictionaries (Cheque and Check), Wikipedia, Grammarist, Writing Explained
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