Are you looking for self hosting or hosting on someone else's server?
The reason that I ask is because if you host it yourself, then you can do pretty much anything you want. If you host on someone else's servers and its free, chances are that they'd want to make some sort of money from your blog so the money from the ads would more than likely go to them.
If you go down the self hosted route, Wordpress tends to be a firm favourite.
If you want someone else to host it you have quite a large choice from Wordpress.com, Posterous, Tumblr, Blogger. However, I'm 99% certain that none of these sites let you make money of adwords, since they are free to start with.
As stated below, blogger allows you to monetize your blog with adsense.