Ok this may seem like kind of a newbie question, but here goes.
I'm using Amazon Web Services and created a Elastic Load Balancer for 2 instances that are identical EBS EC2 instances. We'll call these Server A and Server B. When a request for a file on the main load balancer, I'm assuming it balances the load between and pulls the file from either Server A or Server B. Now if I want to make a change to one of the site files on Server A. Do I then have to make the same change to Server B's file?
Yes, the content on both servers needs to be the same for the sites served through the ELB, otherwise you'll end up with different content being delivered.
You do need the same files on both. One way to achieve that is to use a cluster file system, such as gluster - you can mount the file system, and it will automatically replicate changes between the servers (or if you grow to more servers, you can setup a distributed file system so that you don't end up with too many copies of the same file).
I believe that another approach is possible, but it would largely negate the point of a load balancer. In some reverse proxies it should be possible to setup a redirect if a file doesn't exist, where the redirect points at another server. So, essentially, check for file - serve if exists, otherwise, use the other server. Frankly, I see no point to this approach, in this scenario.