Hi Laura,
I assume your site is using the Office Live Web Design tool vs a 3rd party tool such as FrontPage or Dreamweaver. If so, then this task is fairly simple to accomplish in Office Live to create the registration form. And you'll need SharePoint Designer to create a workflow process to send confirmation emails to customers (This task is not so simple. I recommend you obtain the services of a Web Designer/Developer for this part).
First, you must do two things:
1. Activate the advanced design features (Go to Home>Web Site> Site Actions>Activate Advance Design features to activate).
2. Activate the Office Live Business Applications if not already.
Go to business applications and create a blank workspace or application.
Then create a custom list and add the column fields for your registration form (e.g. first name, last name, email, address, city, state, zipcode). Open SharePoint Designer and create a new workflow that will email users upon submission.
Publish the form to your Public site and your set.
You won't need a database because Office Live Business application already uses a database engine (SQL Server 2005) .
This is just a high level step by step instructions. If you need more details visit Tutorials Live for more information.
Hope this helps!
Norris