You are not going to "move your domain" to AWS, AWS does not offer domain registration services (I hope that someday they will, but they do not)
EDIT: Starting in the summer of 2014, AWS now provides domain hosting services.
Thus, your domain will remain with the domain registrar, this is necessary. Assuming your domain registrar is offering DNS, and assuming that all you want to do is start hosting your website on AWS instead of where it is now, all you have to do is install / specify A record in their AWS instances - they will remain records, MX records presumably will point to your current mail host.
If your current host does not do this for you (because they are a "type of all or nothing"), you still have many options, you can still save your domain with the current registrar and use the AWS 53 route to process your DNS. You set a / cname records to point to your site (presumably what you want on EC2), and you can set MX records to point to the mail server on which you host yourself on AWS (a really bad idea), or an external provider such as gmail or in person, I use Rackspace email hosting.
You need to understand how DNS works much better, and as soon as you do this, you will understand that if you control your DNS, you can have different bits and parts of the same domain, divided into many places / providers, do different parts of what you need.
My typical setup is this:
- Namecheap has my domain
- AWS Route 53 has my DNS (I forward DNS from Namecheap to route 53)
- EC2 hosts my sites
- My incoming POP3 messages are sent to Rackspace
- AWS SES handles all my outgoing emails (just a preference, Rackspace can do this).
Regarding hosting your own mail server on Ec2, this is usually not a good idea - first of all, hosting a mail server is a pain in the neck - leave it to someone who does this all day and pays them about $ 1- $ 2 / month for the privilege . Sending e-mail from an EC2 instance directly is an even more complicated idea - almost everyone will blacklist your e-mail in a short time.
So you have many options - you are not locked anything.