I just downloaded The Free Version of Unity, found the file on my computer clicked it to open it, it asked for an install. So I installed it...nothing much happened. Then I clicked the Unity file again and it asked for an install again.
Why isn't Unity opening?
The thing you downloaded is the installer for the Unity app. After you installed it, then go to 'Start', 'Programs', 'Unity' (I'm not on my developer PC so I can't tell you the directory it's in or the actually name in the program listing).
edit:: I said 'Run' and meant 'Start'. I changed it above.