As these days many people are starting there own Blog, the very first question they ask where should I host my Blog? Especially the one who is concerned about the search engine optimization because when they setup a Blog, they continuously do optimising for there Blog to get traffic. Getting links is one of the important things that many people are concerned about. Once you set up a Blog the juice of links come in and with it you generally increase your ranking at the same time your traffic to website.
Generally there are two ways how you can host your Blog.
1) you can host it on your own website
2) you can host it on free Blog hosting like Blogger.com

If you host your Blog on your website then it will help you in improving your website search engine rankings because the links that you make to your Blog will raise the professed importance of your website.
If you host your Blog somewhere else it will also works fine even though if you had setup on the free Blogging platforms like Blogger.com. I this case also I would gather links to my Blog and do SEO for better rankings which would improve your Blog importance , and this Blog will be linked to my website which in turn will improve its rankings. In this case when someone searches for my site, two sites appear in results page instead of one.
You might be thinking which approach is better? Well you are at the same question again. It all depends on how you think about Blogging. If you want the Blog look and feel to be nice and free enough to maintain the Blog by installing the software etc then go ahead and install wordpress or Movable Type on you website and start Blogging. But if you think of washing hands after setting up and don’t mind how it look like and satisfied enough by using the default styles available with free Blogging platforms then simply host your Blog on free Blogging platform.
When you start Blogging it is most important that you add stuff to your Blog that adds value to you Blog and your website directly or indirectly. Until you maintain good content in your Blog it doesn’t matter where you host to get the juice of Blogging.