The idea is to positively identify you with properly issued government ID. If you show up with your passport, visa, and another form of government-issued identification which the banker can recognize and use (for example - international driver's license, a US-State driver's license, EU internal ID, etc) - it will be quick and painless. Usually, at least two distinct forms of identification are required from foreigners: passport and something else, and not the visa stamped in a passport, that just shoes to show your status upon your W9/W8 requirements may be based. You'll probably be asked for a TIN before any payments are made to you by the bank.
If you don't have anything credible to show as your identification it will be equally quick and painless, except that you'd be leaving without a bank account. If your identity cannot be established properly there and then - they will not serve you.