One definition of an account is a record in the general ledger that is used to collect and store debit and credit amounts. For example, a company will have a Cash account in which every transaction involving cash is recorded. If the company sells merchandise for cash, the Cash account will be debited and the Sales account will be credited.
Another definition of an account is a record of a customer relationship. For example, if a company sells merchandise to a customer on credit, there seller will have an account receivable and the purchaser will have an account payable.
The term on account means not for cash. For example, if a company purchases merchandise with the terms net 30 days, it means the company has 30 days in which to pay.