If you're using Google Chrome, then I know your pain.
Here is why:
"In the fall of 2013 Google announced their plans to discontinue NPAPI support in the Google Chrome browser by the end of 2014.
The NPAPI is the API that makes it possible to run native code in the browser and is what the Unity Web Player is based on.
We are not sure exactly when the NPAPI will be discontinued but expect that Google will stick to their plan.
The long and the short of it is that
when Google do switch off NPAPI the Unity Web Player will no longer work in Chrome."
FEDNET: WOULD YOU LIKE TO KNOW MORE?
A workaround is to use another browser like Safari (I can personally verify that the most recent Mac version of Safari works with Unity) or Firefox.
The good news is…
Unity 5 has just been released.