If you're a little confused, this might help. It's possible to use your AR DSI to backup/restore your saves but it takes a little work.
Download gmtf.dldi from here: http://dldi.drunkencoders.com/index.php?title=Datel_Games_%27n%27_Music_%28microSD%29
Download Pokedoc's savegame manager (that evandixon mentioned) here: http://projectpokemon.org/forums/showthread.php?12405-A-tool-to-backup-HG-SS-saves
Follow the instructions on this page to patch savegame_manager.nds: http://dldi.drunkencoders.com/index.php?title=Win32_GUI
Use this to run an FTP server on your computer: https://sourceforge.net/projects/smallftpd/
I'm not going to go through the process of setting up the FTP server, but it's really easy.
Now modify savegame_manager.ini with the IP address of your computer, user name, password, and port you set for your FTP server and copy it to the microSD card you use in your Action Replay. Turn on the DS with just the AR inserted, go to files and select savegame_manager.nds. The program will run, press B, choose backup or restore, eject the AR and put in your Pokemon game when it asks.