An everyday bank around is one which is designed to have easy access to your money and to make transactions through. As it is an account that is optimised for transacting, banks usually do not pay you interest on the balance (or only pay very low levels of interest).
These accounts usually have a monthly fee which often includes a certain number of physical transactions (from a teller at a bank branch) with fees for additional transactions or services such as issuing a bank cheque.
The account will be able to be accessed physically at a branch or ATM as well as online (internet banking and app). Often a debit card, which can be used for EFTPOS transactions or at an ATM will be included with the account.