Students without a Temporary Residence Permit cannot open an Estonian bank account for free. Account opening fees can range from 100 to 200 euros depending on the bank and your citizenship. This is due to anti-money laundering laws and restrictions on opening a bank account for non-residents.
If you are here on a D-visa and cannot apply for a TRP or are yet to receive it, then you can look into opening a bank account with one of the following online banks or financial service providers: N26, Revolut, Wise. *
Most of these offer free accounts with no monthly fees and give you an European IBAN which allows you to receive and send bank transfers in Estonia (e.g. to receive scholarship payments; pay rent or tuition fees).
We recommend you thoroughly research which one of these options is the most suitable one for you as there may be citizenship and residency-based restrictions.
All TRP holders and EU citizens who have registered themselves as residents in Estonia, should be able to open an Estonian bank account with no issues. Some traditional banks operating in Estonia are: LHV, SEB, Swedbank. *
Some of them have bank cards with deals and benefits aimed at students or young people in specific age groups, so we recommend you look through different banks to find the best option for you.
* The university does not endorse any of these providers over the other and cannot advise you on this matter any further. It is up to you to do the necessary research.