Amazon defines Elastic IP in this way – An Elastic IP address is a public IP address, which is reachable from the Internet. If your instance does not have a public IP 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.
To connect to Instances, from outside the Cloud provider, we connect using the Elastic IP address.
For Amazon, please refer this http://docs.aws.amazon.com/AWSEC2/latest/UserGuide/elastic-ip-addresses-eip.html for Linux and this http://docs.aws.amazon.com/AWSEC2/latest/WindowsGuide/elastic-ip-addresses-eip.html for Windows.
Region support depends on the cloud provider.
Elastic IP can also be added for Azure.