Banks are quite hesitant when it comes to opening accounts for EU/non-British people; out of all the banks I've tried, only HSBC gave me an account and here's what you need to know:
You'll need a proof of address, which has to be preferably a utility bill BUT you can argue with them by giving them a proof of study, a letter from your university which comes to your address with the following information:
Name, date of birth, course start/finish date, course name, your address, and preferably signed by an authority.
You'll need to state how long you've been living in the UK;
Remember this now, if you don't have a utility bill or a driver's license as a proof of address, you're gonna have to tell them you haven't been living in the UK for longer than 6 months, otherwise they won't give you an account. What they'll give you is a basic bank account which you'll be able to upgrade after 6 months to a standard (this is to build up some credit history).
You cannot open a bank account online, because they'll need to verify your information face-to-face so you'll need an appointment. Bring in any information you have - passport, driver's license, any bill or a bank statement that can verify your identity AND address. They're very picky about the latter.
You will receive your debit card by post in 1-3 weeks after the appointment, usually the person let's you know if your interview was successful or not straight away.
Hope that helps
any further questions are always welcome!