Disclaimer: We recently redesigned our Admin Area. Your store may look different than the one shown in this video, but most click paths were not affected.
By default, new customers are required to create an account with your store the first time they place an order. Likewise, current customers are required to log in before they can check out.
If you would like to give customers the option of checking out anonymously, go to Settings > Config Variables in your Admin Area. Select Checkout Variables from the Filter menu and activate the Allow Anonymous Checkouts value.
Now, on the checkout page, the create password and retype password fields are not listed as required for new customers. They still have the option to create an account, although it is no longer required.
Note that the email address field will still be required and a customer account will still be generated in the Customer Accounts table in your Admin Area, but the account will be marked as Customer is Anonymous in the Optional Fields section.
If you do not want your customers to be able to create an account, select All Checkouts Anonymous from the previous Checkout Variables table. This removes the returning customer option and the password fields from the storefront.