Surprisingly, opening a Swiss bank account is not that much different from opening a standard bank account. You have to fill out forms and provide documentation that proves who you are and what you do. However, due to some special circumstances regarding privacy, the level of scrutiny over providing official documentation of your identity is more strict. For example, you may need to show your official passport to provide your identity, whereas a driver's license would probably suffice in the U.S. There are also different minimum balance requirements depending on the type of account you want. These can range from a few thousand dollars to millions of dollars.
The main benefits of Swiss bank accounts include the low levels of financial risk and high levels of privacy they offer. The Swiss economy is one of the most stable in the world, and has not been involved in any conflicts in hundreds of years. Furthermore, Swiss law requires that banks have high capital requirements and strong depositor protection, which practically ensures that any deposits will be safe from financial crisis and conflict.
In terms of privacy, Swiss law prevents the bank from disclosing any information regarding an account (even its existence) without the depositor's permission. The only exception, the only way information will be revealed, is if a government agency makes the claim that a depositor is involved in a serious criminal act or is involved in some other financial issue (such as bankruptcies, divorces and inheritances). Furthermore, Swiss anti-money-laundering regulations also require depositors to provide proof regarding the origin of the funds they are placing in their accounts.
For related reading, see Pros And Cons Of Offshore Investing.