Nope, you don't have to download anything as long as you are hosting stuff on their domain.
Simply go to
www.blogger.com , sign in with your Google (gmail) account. You'll be presented with an admin panel where you'll be able to choose http://<yourname>.blogspot.com subdomain. You can then login to blogger admin panel just like the way you login to Gmail/Yahoo Mail and write posts & publish them.
Similar is the story for wordpress.com.
But if you own a domain and want to setup a blog on it, you may do so by downloading the wordpress software from
WordPress › Blog Tool and Publishing Platform.
I hope this helps. Let me know if you need further help.