Before high-speed internet and app stores, the "Golden Age" of browser gaming was defined by Adobe Flash. Today, while Flash is dead, its spirit lives on through HTML5 and Unity WebGL. Unblocked game sites are the primary preservers of this legacy.
Legitimate browser games do require you to download a ".exe" or ".zip" file to play. If a site prompts you to download something to play Super Mario or Tetris , close the tab immediately. It is likely a virus. unblokedgamesg