Python 2 + Pygame, 178 Bytes
-2 bytes thanks to @ASCII-only, because I was being an idiot
-12 bytes thanks to @officialaimm, also because I was being an idiot
-11 bytes thanks to @JonathanAllan, for telling me the default window size was the screen size
-2 bytes thanks to @WheatWizard, also because I was being an idiot
-18 bytes thanks to @nore, because I didn't have to use the if/else statements
-24 bytes thanks to @nore, who reminded me to put the whole loop on one line.
-42 bytes thanks to @WheatWizard
-3 bytes thanks to @ASCII-only
from pygame.locals import*
from pygame import*
This is definitely no longer the best way to do this, so I may come out with a version that's better suited to the new requirements of the question. The dot is 10 pixels by 10 pixels. The screen size is the size of the computer screen.
If you can see the dot, I'm impressed, as it is almost the exact same colour as the background, to save on byte counts for RGB colours.