1) Allocate an Elastic IP on your AWS EC2 Panel
2) Assign Elastic IP to your AWS Instance
at this point you can reach your server with the assigned IP. Eg 55.544.433.322:8080
3) Switch to the Route 53 Panel in AWS
4) Here you can register a new Domain, note that this domain is note directly hosted by AWS, but it is seemlesly integrated into AWS
After the Domain got activated (after payement) you:
5) Create a "Hosted Zone" again at your AWS Route 53 Panel
6) Usually the correct NS and SOA Records are correctly set-up. If not, create the proper Record Sets for your Region. Check AWS Help on this.
7) Create a new "A-Record", in the "value" field input your Elastic IP from Step 1
If want to create subdomains like www.domain.com or test.domain.com
You crate another A-Record where "name" is the first part of the domain (eg www or test) - the IP is remains the allocated IP.
You could skip step 1 and 2 but assigning an elastic IP to your instance assures that your IP stays the same even if you shutdown your system. Also you can always assign the EIP to any instance without changing the DNS-Records.
If you want your service running on port 8080 to be available without :8080
you have to setup eg. nginx as port-proxy.