The Affero license modifies the GNU Public License in one important way. 

 It has one added requirement: if you run the program on a server and let other users communicate with it there, your server must also allow them to download the source code corresponding to the program that it's running. If what's running there is your modified version of the program, the server's users must get the source code as you modified it.

The purpose of the GNU Affero GPL is to prevent a problem that affects developers of free programs that are often used on servers.

Suppose you develop and release a free program under the ordinary GNU GPL. If developer D modifies the program and releases it, the GPL requires him to distribute his version under the GPL too. Thus, if you get a copy of his version, you are free to incorporate some or all of his changes into your own version.

But suppose the program is mainly useful on servers. When D modifies the program, he might very likely run it on his own server and never release copies. Then you would never get a copy of the source code of his version, so you would never have the chance to include his changes in your version. You may not like that outcome.