At a bank, you can say "I want to open a current account", "I want to open a current deposit account" or "I want to open a checking (cheque) account.". All of these terms mean the same thing. They just have different countries of origin. Any bank employee should understand each of the terms. The use of the word "current" means that you would like the ability to make deposits or withdrawals at will (meaning you do not want to wait to withdraw your money). It also means that you will have acces to your money either physically (at the bank teller window or an ATM) or by check/cheque.

As I know, account is an amount of money you bring to the bank for keeping and increasing little interest during any time (for a month, for a year, etc), and you can withdraw or deposit some money when you want to. As for deposit, this is an amount of money that you bring to the bank for keeping and the latter uses your money at the definite ways. In return the bank pays quite high interest for using your money. But in this case you can't take your money back till the period of time during which you want to profit is over.