I wrote this as a quick summary of the ways I'd recommend to backup your Macintosh.

Use the built-in Apple software, called Time Machine
Use a third party application, like Super Duper!
Use an online, offsite backup solution, like Jungle Disk

When you first connect a blank disk to your Mac, a dialog window will appear and ask if you wish to use the disk for Time Machine. Confirm the use of the disk for backups. If needed, select folders to exclude from backup (to save space). Open the Time Machine options at any time from System Preferences. I recommend turning on the "Show Time Machine status in the menu bar" so you'll have a visual indication that it's running (and may cause other disk-intensive processes to be slower than normal).

The upside of using a third-party backup to make an image of your internal drive is that you don't need to dedicate an entire disk to the procedure, and you can make periodic images of the entire disk and set them on a shelf. If Time Machine flakes out for any reason, you'll have a known state you can return to.

Lastly, you can signup for an online backup solution that is always connected through the internet to their servers. Their client-side application monitors the files on your computer and sends changed files over the wire to be archived. If you ever need to restore them, you just connect to the backup volume and drag them back to your computer.

If you're looking for a hard drive upon which to trust your precious photos and music files, I recommend external drives with the Mac logo by Seagate, LaCie, and G-Technology. There are plenty of cheaper drives, but I've always had good luck with these. Still... it's a good idea to backup in more than one way; just in case.