WordPress Hosted vs Self Hosted
There are 2 Flavours of WordPress. WordPress.com and WordPress.org.
At WordPress.com you can go and signup for a free wordpress blog which will be hosted with WordPress. Typically you will get a blog address that is something like yourblog.wordpress.com there are paid options that allow you to use your own domain name also.
The version of WordPress at WordPress.org is the open source version of the same software that powers WordPress.com, however with this version you get much more flexibility in terms of extending the platform with various Plugins and Themes, furthermore you are able to completely customise the code if more advanced functionality is required.
More explanation of the differences between the 2 can be found here http://en.support.wordpress.com/com-vs-org/