This terminology dates back to the Anglo-Norman Kings who, having conquered Saxon England, started collecting taxes methodically, of which The Domesday Book is a famous example. For accounting, they were using a large board with rows and columns not unlike a chessboard, or un échiquier in French (from Persian origin, imported via Latin).
The person responsible for this task was therefore named Chancellor of the Exchequer. He or she had to verify that the numbers indeed matched expectations. From this exercise also comes the verb to check.
In the meantime, the European banking system started using bills of exchange to credit and debit accounts between branches in various cities on behalf of their customers. Lombard Street in London was famous for this, but they were far from the only ones. This allowed for only paper to be transported, which was both safer and easier.
From 1745 onwards, the Bank of England exclusively had the right of printing money au porteur (payable to the bearer). This happened much later in the US, when the Federal Reserve was instituted. Private bankers who could therefore not emit their own banknotes any more had to resort to a different type of bill that had to be "checked" because they were nominative.
The word check then went back to France, when they imported the concept, and they spelled it chèque and sometimes chècque.
The word then went back again to England as cheque, and only the US actually simplified it back to check.
It should therefore be:
- check in the US
- cheque in the UK, AU, ZA and Canada
As for "checking" vs "chequing" vs "chequeing", my understanding is as follows:
- "Checking" accounts are used in the US,
- both "chequing" and "chequeing" are accepted in Canada with a marked predominance for the former, although the latter is the correct original British English spelling.
- Actually, in the UK you don't hold "chequeing" accounts, but "cheque" accounts, but this phrase is seldom used; "current account" (as opposed to savings account") is preferred. However, when the gerund is used it is spelled "chequeing" (this is disputed however, see Brian's comments below).
- I'm told that in Australia, "cheque account" is used and "chequeing" is rare (comments below from staticsan).
Another famous "ping-pong"-type etymology between French and English is
"taster" (to grope for testing purposes) => "to taste" => "tester" (to try) =>