I'd suggest WordPress - it is the cream of the crop. You can install a standalone version in a directory on your own server (
www.yoursite.com/blog or blog.yoursite.com) or you can use the WP hosted service and be up and running in a few minutes (you.wordpress.com).
Both are free; you will have fewer options to customize on the hosted version than a standalone version.
Download the software at:
http://www.wordpress.org
Get your hosted blog at:
http://www.wordpress.com
The features and ease of use are far better than Blogger. The ability to customize is incomparable. The support community for WP is terrific too.