Follow the directions in a previous post on OzBargain (http://www.ozbargain.com.au/node/42072) to get the Amazon appstore on your phone. It works, but it takes up to an hour.
The free app for today is the newly released Plants Vs. Zombies that hasn't yet made it to the Android Market, and may in fact be an exclusive to Amazon for a time anyway. The game plays beautifully even on an Android phone.
To get this when it's not free may even be tricky if using a fake American identity.
Edit: As noted by wreckedandjealous below, once you set up the Amazon App Store you can get a free app every day. In the last week I have downloaded a few cool games and fitness apps. If you don't like them you can uninstall and they remain in your 'purchased' list if you ever want to download them free again later on.
Mod: Change the URL to point to the product, but you still need to use the above link to get access to Amazon App Store.
Edit 2: If you're uncomfortable with the process above, hang tight. According to sources on the web, this will be available as a paid item on the Android Market in two weeks or so, which is when Amazon's exclusive runs out. See http://www.androidpolice.com/2011/05/16/begun-the-market-war… and http://www.pocketgamer.co.uk/r/Android/Plants+vs+Zombies/new… for example…
should also note that while it does take an hour, you are set up for their free app every day… just log in and download