Try copying the contents of the DVD to the computer's hard drive, and playing it there.
If the copy fails or takes a really long time, either your DVD burn was marginal, or your DVD drive is marginal or failing. The DVD player is able to scrape enough data off of it so that the error correction gives the correct video stream, but the computer cannot. If you suspect it's the former, you can try burning it again, maybe try different DVD-R blanks. If the problem is the latter, you'll need to replace the computer's DVD drive.
You can use Nero DiscSpeed to test the quality of a DVD burn (though you really need to test the same DVD on multiple DVD drives). You may wish to find Nero DiscSpeed version 4.x.x.x. I hear they added OpenCandy spyware to the newer versions:
http://www.majorgeeks.com/files/details/nero_discspeed.html
If the copy works but the video still fails to play, you probably need to update the video codecs on your computer. I've had good success with both K-Lite and CCCP.
A very rare possibility is that your DVD-making program is using a non-standard DVD format which the DVD player understands or is ignoring, but your computer is getting upset about it and stopping/crashing.