Some different blogging platforms:

If you want to set up your own on your own server or server space that you lease (hosting account):

• Wordpress
  This is my personal favorite (and what this blog is running on). Very easy to install and customize. Take about 5 minute go from no blog to blogging. A very large support community with tons of plugins and themes. Requires PHP and MySQL.
• Serendipity
  Another great opensource blog. I used this one in it’s early days. Pretty easy to install and use. I have a buddy that uses it on his blog and he really likes it. Looks like the Serendipity Community is also growing a bit.
• LifeType
  Now I have not uses LifeType in a long while. The project use to be called pLog and it had to change because of now trademark problem or something. It has seen some recent development but the user community is not that large. Still worth looking into though.

Hosted:

• Wordpress.com
  Allows you to start your own Wordpress Weblog. Free to sign up for and very easy to use.
• Blogger
  If you are technically challenged or you are just looking at getting started blogging then check out Blogger. My daughter has her blog here and she has fun with it. It is also very easy to use.
• There are hundreds more. Just do a search for free blogging.