Banks provide some free services for their customers, like cashing checks. You ask the bank teller to cash a check, so she needs to know if you are a customer of that bank. She asks you:
Do you have an account with us?
When you "have an account" with a bank or a company, it means that there has been a customer relationship set up between you and the company. You and the company are able to pay money back and forth through this "account". Your bank account is one example. When you put money into your bank account, the bank keeps a record of how much money it owes you.
Other kinds of companies that you might "have an account with" include your phone company, your electric company, a credit card company, and so on.
When you want to express which company your account belongs to, you use "with":
Do you have an account with Citibank?
I cancelled my account with Time Warner Cable.
The word "account" is now also used to describe services that you use on the Internet. Examples of this is your email account, Twitter account, etc.
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