This guide assumes that you know a few basic things about AWS, that you have logged in and created an instance, and can navigate the AWS menus. If you have not created and instance, check this out: Creating an Amazon AWS EC2 Instance
An Elastic IP address is a static IPv4 address designed for dynamic cloud computing. An Elastic IP address is associated with your AWS account. With an Elastic IP address, you can mask the failure of an instance or software by rapidly remapping the address to another instance in your account. An Elastic IP address is a public IPv4 address, which is reachable from the Internet. If your instance does not have a public IPv4 address, you can associate an Elastic IP address with your instance to enable communication with the Internet; for example, to connect to your instance from your local computer.
If your instance is running, it will currently have a dynamic IP Address assigned by Amazon.
If your instance is running, you have to stop the instance to allow the DHCP address to return to the Amazon pool, and to allow assignment of an Elastic IP Address.
Navigate to the AWS Console.
Select the Elastic IPs Option.
The next page shows your allocated Elastic IPs (you probably won't have any.)
Select Allocate New Address.
The next page seems a bit unnecessary.
Next you will see a confirmation page.
Here you can see the Public Elastic IP Address you have been allocated (this is not associated with your instance yet.) Select Close.
Once back at the EC2 Console. You need to start your instance.
Select Actions, Instance State then Start.
You will get a prompt asking if you wish to start the instance.
Click Yes, Start.
Back at the EC2 Console you will see your started instance.
The IP Address should match the Elastic IP Address you were allocated when you created it earlier.
That is it, pretty simple really. You can now connect to your AWS EC2 Instance using your new Elastic IP Address.