The game uses HTML5 and you can also try it on your smartphone or tablet. You can share your score on Google+ and generate URLs that remember your high score.
Here's a video that shows the game, just in case you don't see the doodle yet (you can also visit Google Australia):
The Google Doodles site has a list of all the other birthday doodles: the first one is from 2002. The doodle from 2007 has a lot in common with this year's doodle.
Happy Birthday, Google!