WordPress.com

WordPress is by far the most popular blogging platform. Its hosted platform is very good too and provide quite a lot of tools for free. You’ll need to pay if you want to be hosted by them and have your own url, but it’s still quite cheap. Some further customization can be done after payment.

Blogger

Before the rise of WordPress, Google’s blogging platform was the most popular. Setting up your blog there will be extremely easy and will allow you to have your blog linked to your Google account, which can be quite convenient.

Free web hosting with free software

Software

There is tons of open source blog engines out there now, WordPress being the main one now. You could also give a try to TextPattern or Drupal.

Free web hosts

Doteasy.com
Doteasy is probably one of the best free web hosts out there. No advertising, free email accounts and website creators.

Paid web hosting with free software

You can still use the open-source mentionned above while hosting on a paid web host, this will be more reliable but will give you a little more work.
Dreamhost
Dreamhost is a company ran by a great team of software enthousiasts who will always be there to help, I strongly recommend them.
HostGator
HostGator also has pretty much the same resources than Dreamhost, with in my opinion a little better control panel.
FatCow
FatCow is a younger company, but they have a really reliable and extremely affordable service.