Hmm...it could be the Windows Media Player codec you are using on the videos you are trying to open.
i guess when you say burning a DVD disc, you mean copying data files and not a movie onto it?
IF ITS A MOVIE:
If its a movie, it would have to be copied as a DVD using some software and not just copied onto the DVD player by dragging it on, It'd have to be set up, since it'd convert the movie to something called MPEG-2 to play on DVD players. In which case, use some software called Nero which you might have got with your DVD copier and select to make a DVD video or something along those lines.
IF ITS JUST DATA, AND HAVE COPIED JUST STANDARD MOVIES TO PLAY ON PC ONLY:
In this case your not using the right codec to view, and your computer has a codec to de-compress the movies on disc that your other computer hasn't got.
What program are you using to make the movies to copy onto the disc? Try using a different codec in this case.
A codec by the way is a compression format which is burnt within the movie to compress it down to a smaller size, since a uncompressed movie wouldn't fit onto a DVD, which is why DVD 's in shops use a codec called MPEG-2 which is a high quality compressor, and to de-compress this data back to play it, the computer needs to support it in the first place.
Try using more standard windows codecs that the other computer shouldn't have a problem with
Hope this long list helps