A billable Account is any entry in the Address Book that can be billed against. To setup an address to be a billable account simply set the Billable Account field, found in the Billing tab for the Address Book entry, to Yes. Optionally you can also set the category of account.
Only when you have set this can you assign a case to the account, and setup the account for its own rate table.
For billable accounts, the Contact screen has a different meaning:
The Name field is your internal name for the account, and the second name line reflects the actual name to be used in the invoices.
Next: Setting up the account