There are some games on the market which are run directly from a website. You have to log in there and press play. Then the process is created on your computer and you can normally play it as if it was run from .exe. It actually is run from .exe but the game opens only if I load it through website. Running the .exe alone won't open the game. How is it all done? Is the .exe somehow packed and the site unpacks it somehow? I would be very grateful for some brief explanation or articles!
Check this one out: http://3d.wolfenstein.com/game_NA.php
Drawing divs is actually fast. It's where most browsers focus because that's what they do.
Then there's the HTML5 Canvas element, which is also being used more and more places (albeit not as many places as regular html elements)
EDIT: And while i wrote the above you noted that the games you mentioned aren't browser-based. In that case, you'll need to "Run" an executable, a thin client or something similar, which is then actually downloaded and then executed. (And in that case, you were right in that's an exe)
If the game runs within the browser, it's not an executable, but an interpreted program. (Either flash a java applet or silverlight, as frob mentioned)