I'm trying to help a friend set up a projector to watch movies.  It
works fine with my laptop.  Hers appears to work fine until you
actually start to play the DVD.  That is, anything else that shows up
on the screen projects normally, but inside the DVD player window,
nothing is projected; it just looks black, even though the movie is
playing on the laptop screen.  This is true regardless of the software
used to play the DVD (Real, Windows Media, WinDVD). The projector
connects to the computer using the 15-pin VGA port. I can give details
about the projector if necessary, but I think the problem must be on
the computer end, since it works with my laptop.  Both computers run
Windows XP.  Is there some setting that controls what gets sent to the
VGA port or something?

try turning off hardware acceleration on the graphics card. to do this
with Windows Media Player 9, Click Tools, Options... Preformance, then
slide the Video Acceleration to none. There might be a box that pops
up, but just hit ok. Then just restart the player.

You need to change the default monitor. Right click on the desktop, go
to properties, monitor tab, change default monitor and try again. You
might also want to try VLC player, move it to half the screen then
slowly move it over to the second screen. This should work, please let
me know if it doesn't..

Here is what finally worked:

Display-->Settings-->Advanced-->Overlay Controls --> Video Mirror
Controls--> Change from Disable to either Primary or Secondary Display
(under fullscreen device). These steps are quite specific to this
model, but searching the online manual with the phrase "disable
internal LCD display" netted the necessary results.