Commerce stores addresses for customers automatically. When users are logged in during the checkout, it will also associate the addresses with a user record, and allow the customer to choose from previously addresses on their next purchase.
Here's an example of how that could look:
There is no additional setup needed to use this functionality. It's built into the commerce core and default templates.
Setup (customer section)
The architecture makes it possible for address-book-like functionality for logged in users to be added to Commerce, but that's not currently (as of v0.10) available.
Under the hood
comAddress has a
user field, which is either 0 or the ID of a user. It also has a
remember field - if that is enabled, the address will show up in future checkouts.
Addresses are joined to an order through the
comOrderAddress object, which has the order and address ID, as well as a
type field that dictates if it is a billing or shipping address.