Host website on AWS using existing domain name registered at Register.com

I'm going to assume that your domain is hosted by register.com rather than Route 53.

For the most part, who hosts the domain (that is, who is responsible for the servers that respond to the DNS queries) isn't particularly important and doesn't have to be related to where your website is being hosted. So if your DNS is currently hosted at register.com you can use that without using Route 53.

You don't say how you're hosting your website in AWS but in general, it's going to be as easy as creating an A record in your DNS and pointing it at the IP address of the EC2 instance that is hosting your website; or creating a CNAME record and pointing that to the name of the load balancer that you're using. The instructions for Route 53 will be much the same.

The edge case here is if you want to use the "apex record" in your domain. Say you own example.com - it's straightforward to create a record for www.example.com and have that point to an EC2 instance or load balancer as above. But if you want to use example.com (the domain itself) as a record then that's a little tricky and is generally easier if your domain is hosted in Route 53.