In Germany we use a current/checking account (Girokonto) provided by a local bank such as "Sparkasse" or a national bank such as "Deutsche Bank". Most banks offer a Girokonto without fees for students.
When opening a bank account you receive an EC card (electronic cash and debit card) with a PIN (personal identification number). You can use the card for cashless payments or withdrawing money from the cash machine (ATM). Other banks may charge extra for withdrawing money from their ATMs. To print out your bank statements, use the printer at the bank, which is usually right next to the ATM.
Nowadays, you can easily get online or telephone banking for your account if you ask for it. Every bank account comes with 3 numbers, the account number (Konto-Nummer), the routing number (Bankleitzahl) and an international bank account number (IBAN). You should know these numbers for international money transfers. For regular expenses such as rent and health insurance, you might be required to give permission for automatic withdrawal. Make sure your account always has enough money for these regular expenses.
An EC card or cash are the main methods of payment in Germany. Traveler's cheques are not very common in Germany. They are either not accepted or charged a high fee. In most cases, you are better off using your EC card.