Summarize the following:
In order to install a new hard drive and back up your games to it, you'll need to mod your Xbox console. This will require an Action Replay so that you can copy files from your computer to the Xbox and an original copy of a game that can be exploited to install the custom software. The three most common games used to mod the Xbox are "Mech Assault", "007: Agent Under Fire", and "Splinter Cell". For detailed instructions on modding your Xbox, click here.  Download the exploit for the game that you are using. You can find these exploits on a variety of different Xbox modding sites. Copy the exploit files from your computer to the Xbox memory card, and then copy them from the card to the Xbox HDD. Start the exploitable game and load the save file that you copied. This will begin installing the mod software. Select the menu option to start the installation. The menu will vary depending on your exploit. You'll usually be asked to choose a new dashboard, which is a personal preference as they share functionality. The standard hard drive on the Xbox will not be large enough to store more than one game (if that). Replacing the hard drive allows you to install a massive hard drive that can fit hundreds of games. Open your Xbox and remove the old drive. You'll need a Torx 20 and a Torx 10 screwdriver in order to access the hard drive. Click here for detailed instructions on opening the Xbox case and removing the hard drive. Make sure to store the drive in case you need to use it to restore the Xbox. This is a program for your modded Xbox that can copy your game discs onto your Xbox hard drive. You can then play these backups directly from your hard drive. Some exploits may have installed this program automatically, or you can use the Slayer Auto Install tool to add it to your modded dashboard. You'll need to insert the original disc for the game that you want to backup to your hard drive. You'll find this in the "Applications" section of your Xbox's dashboard. You'll be asked to select a directory. Choose the "e:\games\" folder to store the game. DVD2Xbox will automatically create a folder for the specific game. This will likely take several minutes to complete. You'll see your newly backed up game listed here, along with any other games you've backed up. Selecting a game will launch it without the need for the disc.

Summary:
Mod your original Xbox. Install a new hard drive (optional). Install DVD2Xbox. Insert the game that you want to backup. Launch DVD2Xbox. Select "Copy DVD/CD to Harddisk". Wait for the game to copy. Return to your dashboard and select the "Games" section.