With Amazon's EC2, to assign an IP address I create an "Elastic IP", however it seems to just give me a static IP address. What is the difference between a static IP and an elastic IP? Or is it the same and just part of AWS's branding?
Amazon Guides are always a good reference to understand better the entire infraestructure. In Amazon EC2 Elastic IP Addresses you can read:
Elastic IP addresses are static IP addresses designed for dynamic cloud computing. Unlike traditional static IP addresses, however, Elastic IP addresses allow you to mask instance or availability zone failures by programmatically remapping your public IP addresses to any instance associated with your account. Rather than waiting on a data technician to reconfigure or replace your host, or waiting for DNS to propagate to all of your customers, Amazon EC2 enables you to engineer around problems with your instance or software by programmatically remapping your Elastic IP address to a replacement instance.