Wordpress is definitely the current hot blogging platform. It is very easy to install on your server, has a huge library of plugins and adding new themes is a doddle. It is free and open-source too, meaning it is constantly improving and the updates are free.
Other software options include Movable Type and Expression Engine.
If you don't want to host your own software you can also use a web-based solution. Blogger.com, Wordpress.com, Typepad.com are all excellent web based blog apps but they each have a slight variation on the theme.
For example, blogger.com allows you to add Google ads, Typepad is marketed toward being extremely easy to use and Wordpress.com has the advantage of getting you familiar with the admin console in case you move to their popular self-hosted solution in the future.