The taskbar buttons on my Windows XP taskbar have mysteriously
disappeared, so I have no idea what programs I have open.  How can I
display the windows buttons in my Windows XP taskbar?

In order to cycle different windows/applications use the following keystroke:

Alt + Tab

If you hold down the Alt key after pressing Alt+tab, Windows pops up a
box in the middle of the screen that shows you what is open.

Another way to see what is open you can press the following keystroke:

Ctrl + Alt + Delete

Windows Task Manager will pop up and you'll have access to a whole
slew to things. For your purposes, click on the "Applications" tab and
you can see which applications are loaded. In order to bring that
window to the forefront, select it then press the button titled
"Switch To" and the window should come to the forefront. Also, if a
program is frozen this will be shown under the Status heading and a
comment saying "Not Responding" will be present.

Once you've finished up what you're doing, you should restart the
computer and most likely your problem will disappear.

Probably what happened was Explorer crashed and reloaded. Sometimes
when this happens, you'll be unable to see which programs are open in
the Windows XP taskbar.

Press Ctrl+Alt+Del to call the Task Manager out. Then click menu
File->New Task, and enter "explorer" to start the explorer.

I had the same issue. Here is the way to solve it. 
Go to: http://www.kellys-korner-xp.com/xp_tweaks.htm
Go to:  #240
Click on the item and download the vb script
Run the script