Because Windows 8 now ships on a vast variety of hardware that doesn't always have a disk drive (e.g. touch devices), Microsoft decided not to include DVD codec in their new operating system to make it cheaper. This leads to the problem that devices that actually do have a disk drive cannot play DVDs out of the box:
The easiest way to circumvent this problem is to install a third-party multimedia player that is able to play DVDs - for example the popular VideoLAN player (VLC Player) which is able to play nearly any multimedia format.
You can download it here.
After downloading the installer, double-click it and follow the instructions. Afterwards you can right-click the DVD again and open it with VLC: