You're not the root user; you are a user which has the privilege to become root, using the
sudo (or various graphical surrogates) method.
In Ubuntu (and in most not-embedded Unix environment) you normally work, and have the privileges, of a normal user. Only deliberately you can override the normal rules; it is a way of telling you that you have to know what you are doing. It is a great safety net that works normally quite well(1).
For example, although it can have legitimate uses, trying to add manually a file to
/usr/bin is normally wrong.
You can find all the info you need here: https://help.ubuntu.com/community/RootSudo
(1) although remember that Unix, in general, gives you rope(2) if you really want to hang yourself.
(2) You can run the file manager application as
root, as explained in the other answer.
Alas, you shouldn't.
This is the rope I mentioned in (1). A wrong click, a spurious double click, and you can have a completely messed up system, to the point of having to reinstall it. I am a power user, and I have to use root powers maybe... once a week? And most of the time is just to install some program, which you can do easily via the graphical application.
You should be able to do all of your work in normal mode; copy and paste between places you own works perfectly with the graphical interface. Why and what are you trying to copy to