1) On their server (wordpress.com)
2) On your server (yourdomain.com)
To put it bluntly, you don't want to have your blog hosted on Wordpress's server. The main reason? Because Wordpress does not allow you to advertise products on a blog that is advertised on their server.
This is a big disadvantage unless you just want to get traffic and send it to your website.
On Your Server: This is the way to go!!!
I highly recommended that you setup a blog on your own server for several reasons.
First... You can advertise whatever you want!!!
Second... By having your own domain name such as (yourdomain.com) it allows you to brand yourself. This helps you to build your name and brand. If you placed your blog on Wordpress's server, your blog's address would be something like (yourblog.wordpress.com) And that's just not very professional.
Third... By placing Wordpress on your own server, it is completely customizable… you can upload different theme templates, upload lots of plugins, and you have the ability to change the code if you know anything it.
More... And just so you know, Wordpress is easy enough for anyone to use. With Wordpress you can actually have your blog setup and start posting in less than half a day.
With all that said… I highly recommend that you setup Wordpress on your own domain. The search engines like Google, absolutely love Wordpress's blogs and many people find it fairly easy to show up on the first page at google for competitive keywords in many cases.
To setup Wordpress on your own server there are a few steps… I will lay this out in a few sections to make it easy for you to follow.
Wordpress Website / Blog Tutorials:
1. How to build a website with wordpress
2. Choosing a domain name
3. Setting up a hosting account
4. Linking your Godaddy domain to your hosting
5. Installing your Wordpress blog
6. Making changes to your Wordpress blog