It's almost impossible to get by without a bank account in Germany. Your wages are paid into it, you pay bills (your rent, for example) with it – these are common transactions for which you need a Girokonto (current account). To begin with, you should look on the internet or find out from friends which banks are in your local area and what they offer.
Then you need to present your ID or passport at your chosen bank. You also need to show proof that you are registered in Germany – for example, a form from the residents' registration office or evidence that you are an asylum seeker or you have temporary resident's status.
What kind of account would you like to open? Do you need a Girocard (current account card) or a Kreditkarte (credit card)? Do you want to use online banking? Depending on the bank and your needs, the fees vary. Ask for information and compare different banks before you decide on one. Sometimes you can also open an account online.