I wrote this content some time ago. For new Mac users, I recommend using Time Machine. The following steps are really only applicable for older Macs or if you want to archive files for later use (clear off your on board hard drive)
To do this you need an external drive of some kind. A USB flash drive will be fine for most users who mainly work with documents such as Word and PowerPoint. If you have saved a lot of photos, video, or music on your Mac, you might want to invest in an external HD. At the time of this writing, you can find 8 GB flash drives easily for under $20, and 500 GB external hard drives for under $100.
Steps to back up your important files. (for Mac OS 10.4 users, for 10.5 or higher click here)
1. Connect your drive to one of your computer's USB ports.
2. You should see it pop up on your desktop. Double click on it to open a finder window viewing its contents.
3. In the menu bar at the top of the screen chose New Folder. An untitled folder will appear within the finder window showing your drive's contents. Name it Backup and the date. ("Backup 10-02-03" this example signifying Feb. 3, 2010).
4. Choose File > New Finder Window, then choose Go > Home. You should now have a finder window open that has several folders in it: Desktop, Documents, Library, etc.
5. Most of us keep our main working files in Documents and Desktop. Holding down the Option key, drag and drop these folders into your new folder called "Backup..."
6. You can know you are making a copy on your drive if you see the green circle with a plus and also see a "Copying files" window pop up briefly with a status bar showing when the task completes. The more files you are copying the longer that window will last. If it is only a few small files it will only flash briefly.
7. If you have saved photos, music, or videos on your computer and bought the larger external hard drive as mentioned above, you should go ahead and copy over all the folders in your home folder, i.e., Music, Pictures, etc. Depending on how much you've saved on your computer, this could take a while.
Notes
If you follow these steps once a week, sooner or later you will fill up the drive. The computer will throw a warning along the lines of there is not enough free space to copy all the files. Click Stop. To free up space: simply trash the oldest backup folder on your flash drive, empty the trash, and try the copying step (#5) again.
Consider where you will store your backup drive. If you keep your laptop and backup drive in the same location, you are guarding against a hard drive failure. If you keep your backup drive in a different location you are also taking steps to guard against theft, fire, loss, etc.
If your files load is not very large, you might want to consider an online solution such as Dropbox. I've been very happy with this solution. This linkwill allow you to sign up as a referral from me; both of us will receive 250 MB extra storage on top of the 2 GB you get for free. If you'd prefer you can simply go to www.dropbox.com and sign up as well.
If you want to know more about backing up your 10.4 system, click here.