For this to work you'd need a couple small changes. As mentioned, use \ to escape the space in GPU Test. If you use tab completion the shell will do this for you.
mv user/me/Documents/documentname user/me/Desktop/GPU Test
When moving a file to a certain folder, always use a trailing /. Here's what I mean:
Reason being that mv can also function as a rename command, so in your original command above the shell would move documentname to the Desktop folder and rename it to GPU Test. The trailing / tells the shell to move the file to the existing folder GPU Test and nothing more. Hope that's clear.
mv user/me/Documents/documentname user/me/Desktop/GPU\ Test/
There's a free PDF here, it's a good tutorial on learning the command line and I think you would benefit by taking a look at it. I've been using Linux for six years and I still refer to it quite a bit myself.