I've had a number of friends recently ask me about the best way to go about transferring a DVD collection to digital format. Since I've got more experience working with a Mac, and all the requests I've gotten are from fellow Mac users, I'll focus on them for this post. PC and Linux users, no hard feelings.
The simplest way I've found to digitalize DVDs is with Handbrake. After you've downloaded and installed it to your Applications folder, pop in a DVD and open Handbrake. It should automatically drop down the source selection window (if not, just click "Source" in the upper left). Select the disc image of the DVD and then give it a few minutes to scan the source. When it is ready to rip, the start button at the top will turn green. Browse for a tasty location to save your video (might I recommend your Movie folder?). Halfway down are all the video and audio preset tabs. These come in handy if you want to alter the output format (say, if you wanted to pull the movie into iTunes to watch on your phone or iPod). However, the defaults should work just fine if you're looking to just be able to watch a movie on your computer. Click "Start" and then go grab a snack. Typically, DVD rips take between 45 minutes and 1.5 hours to finish, depending on the length of the movie and the quality of the rip.
Once Handbrake has finished with your rip, you'll have a nice digital copy of your movie that you can backup on an external hard drive and take with you anywhere without worrying about the hassles of physical media (dirt, scratches, etc.).
If you're looking for other ways to rip your movies to your Mac, I might recommend using RipIt, Mac the Ripper, or VLC (which is also a kick-ass open-source media player). The first two are fairly simple to use. A nice walk-through for using VLC as a ripper can be found here.
For more handy Mac freeware, check out 7 Free Gems for Your Mac.