First I followed the instructions from the Ubuntu Help Center, which you can access from the "System --> Help and Support" menu selection on the task panel. If you follow the link "Music, Videos and Photos" you find a link to "Play movies and videos". There it will instruct you to install the packages gxine, libdvdcss2, libdvdnav4, libdvdplay0 and libdvdead3. You can do that using the "Synaptic Package Manager" under "System --> Administration", where you would use the "search" function to find and install these packages.
The only problem was with the libdvdcss2 package. It couldn't find it in the installed repositories. But you can find instructions on how to install it from the "medibuntu" repository here. You need to execute the two commands you find there by copying and pasting them to a terminal window, which you can find under "Applications --> Accessories". Make sure you exit the Synaptic Package Manager before executing these commands.
After this step I inserted the DVD I wanted to see, and it played like a charm using the "Totem" movie player. Then I made the mistake to continue to follow the instructions from the Ubuntu Help Center, where it says to run the command sudo /usr/share/doc/libdvdread3/install-css.sh, and set the gxine as the auto-play program. After that the DVD wouldn't play in either of the two players, and I had to re-install the packages above to get it working again.
What is your experience with playing DVDs on Linux machines?