Pretty basic scenario. I have installed Joomla! with fantastico and it doesn't want to install to the root directory so you simply make a folder for it to go into. This is great because I can test without it being live.
Now, what files do I need to delete and save and move so I can turn off the redirect which is pointing to the Joomla! install?
Just move all the stuff from inside the folder you installed in to the root, then edit the joomla configuration.php and remove the folder name from the path. You'll find it's in 3 different places inside configuration.php. That's it.
I just recently renamed my install directory from "content" to "web" in order to get SEF working. I prefer to keep it in a suitably named install folder instead of root because it keeps things tidier, I know what stuff belongs to joomla and what stuff belongs to something else. Easier to keep track of things and backup, etc.