There are several options, depending on what else you want the computer to do, and how nice an interface you want.
Right now I use Theatertek as my DVD player. It has the ability to play DVDs stored on the hard drive. Its file menu is nothing spectacular, basically similar to Window's Open File dialog box but easier to navigate with a remote control or a keyboard. No fancy display of the cover art or information about the DVD. However, it is functional, and the ease of use and picture quality of TT more than make up for it.
I also use SageTV (or rather now it is called Sage media center). It can also play back DVD on the hard drive, and can have cover art and other things. However, I haven't tried playing DVDs with it because I have TT. (However, Sage really is a kick-ass PVR).
Other options include Xlobby, which is a freeware front-end that can do the fancy cover art and information, then send the files to an external DVD program like TT. You can even use programs like CQC, which is an Uber front-end/home automation program, and there are other options which I have not personally tried.