Apple - What's the menu area near the clock called?

These are called menu extras.

They are different from status bar items (which are usually used in other apps) in that you can hold command and drag them around.

Each of these can be turned on and off with a checkbox in System Preferences (the Bluetooth one can be toggled in Bluetooth preferences, etc.)

The answer above is true. To add to that:

They're also known as menu items. You can also remove them by holding down the command key and dragging them out of the menu and it will release its magic smoke be removed from the menu bar. (Just like you can remove items from the dock).

It seems that you can only remove Apple menu extras using the method mentioned above. To remove any third party menus you will have to hunt around the file system and remove it manually or get lucky with an uninstaller that removes the program and it's associated menu extra.

To reload the menu extras (which I believe is the correct Apple name for them as per answer above), you can just restart SystemUIServer using /Applications/Utilites/Activity Monitor making sure you have View > All Processes selected in the view menu.

For advanced, you can just do:

In Terminal:

killall -KILL SystemUIServer

to check to see if you have got rid of that pesky menu extra that you wanted to get rid of. If its still running, keep hunting!