In the Republic of Ireland banks are:
All of them are purposing Internet banking.
To open an account you will need a proof of address (bills for example), your PPS number (Personal Public Service number / Social security number), a contract job and a valid Id card. You should try to open an account with a bank that has a local branch. Actually, you will probably need to visit the bank to transer money to an international account for example.
Ulster Bank, Bank Of Ireland, Permanent TSB and National Irish Bank have current accounts with no charge.
For all Lazer Cards there's an annual government tax of €20 per year.