WordPress web hosting works in two different ways. They have a free blog service and then they have user hosted blogging. Here we will go over both ways to have WordPress work for your needs.
If you go to wordpress.com you can set up a free account and instantly start blogging. They have a user friendly interface and fun themes and backgrounds to personalize your blogs. You can also be allowed to post on other blogs if you are registered and the owner of the blog adds you as a user. While the templates are good and fun you have to upgrade and pay a yearly fee to be able to alter them to fit your needs or just to tweak them. Typically you get 3GB of storage but if you upgrade you can have 5GB or more with the yearly upgrades. The free version can contain text ads from Google or other companies and if you want those to be gone you have to pay a yearly fee. The other benefit of upgrading is that you are not limited to the 35 user cap on your blog. If you have many people that you want working on the blog or having the ability to use it for anything other than reading this is what you want!
With your free account you get a subdomain like freewebspace1.wordpress.com. If you register a domain name that you like you can redirect your WordPress blog to go straight to your URL of choice. WordPress also has tools to help you import other blogs that you may want to consolidate.
After you get web hosting that you are sure supports WordPress hosting you can go to wordpress.org and sign up for even more features than you gets at wordpress.com. WordPress offers so many tools for the blogger. They also have thousands of plugins with more being added all the time.
Before you sign up for anything be sure to read the terms of service, consider what your goals for your blog are and what your budget is. This is the only way to really know if WordPress has web hosting solutions for your blog.