According to Swiss law, non-residents of Switzerland who would like to open a Swiss bank account must be at least 18 years old. Other than that, there aren't a lot of restrictions. Your account can be in almost any currency, and there is often no minimum balance required to open an account. Once you've started making deposits, however, there is a minimum balance you have to maintain that varies from bank to bank and by type of account.
If you maintain your account in Swiss Francs you will earn a small amount of interest, but will then have to pay the Swiss withholding tax. Because of this, most account holders that don't live in Switzerland have their Swiss bank account in some other currency, such as the U.S. Dollar, British Pound or Euro, where your money can be invested in a money market fund and earn tax-free interest.
While it's usually better to open your account in person, there are many Swiss banks that will allow you to open an account by mail or fax. There are also many firms that exist to assist people in setting up offshore accounts. Because Swiss anti-money-laundering law requires you prove where your money is coming from, many certified documents are required in order to open an account, including authenticated copies of your passport; documents explaining what you do for a living such as tax returns, company documents, professional licenses, etc.; proof of where the money you are depositing comes from, such as a contract from the sale of a business or house; and all of the typical personal information about yourself such as your birth date, a utility bill to prove your residence, all contact information, and, of course, your name. They'll also want to know what you want to do with the money once you have the account.
You can close your Swiss bank account at any time with no restrictions or cost. You can get your money immediately, and any invested money as soon as it is liquidated.