Right bottom corner textbox in desktop; what's it for?

You mean this?

A similar text box appears in the bottom right-hand corner of a Nautilus window if you start typing in it. There it allows you to select files and folders by typing out the first part of their names. Since Nautilus handles files on your desktop, I'd imagine it does the same thing there.

The desktop is managed by the file browser called Nautilus, which allows you to type in the window to quickly find a file. When you do this, a little box pops up with what you have typed. The same thing is happening on the desktop. Hope that helped?

edit: Looks like WarriorIng64 beat me to it, so in that case I'll just add that I think this may be a bug - it never used to happen, and personally I think it just looks messy. Perhaps it was left there by accident.