I have a Dell Dimension 8100 running Windows XP, SP3. I have 512 MB RAM. When I click the button to analyze or defrag my C drive, nothing happens. When I check Task Manager it indicates that it is running. What can I do. I have run Chkdsk and found no major problems.

Try to defrag another disk attached to you're pc. If you dont have one than borrow one so long. By doing this you can determine if its a windows or disk prob.

Umm... defrag runs under [1] mmc.exe, and employs...
[2] dfrg.msc
[3] dfrgres.dll
and [4] either dfrgfat.exe or dfrgntfs.exe, whichever is relevant to your file system.
You might copy in the four files to system32, or at least check that they exist, and try again.
Then to do a more complete job, navigate to Windows\inf\dfrg.inf ... rclick that file and choose Install.
This last step will tizzy-up your registry entries.

It takes a loooooooooooooong time for it to defrag a hard drive -- expect it to work for a couple hours depending on the size of the hard drive and how full it is. If the hard drive is nearly full then defrag won't be able to do much with it because it needs lots of space to move the data around.