Rosie,
I'll explain it as easily as I can.
-You need a hosting account to store your website
-You need your own domain (business website url)to brand your business and for
SEO purposes
-Wordpress.com is not what you want
You need to decide on your company name then buy a domain that matches as best as possible. Whatever company you buy the domain through you can buy hosting as well. Once you have your hosting account you will be able to install Wordpress onto your new domain (most hosts have a quick install feature for it) then you can go from there.
Hope that clears it up somewhat.
~Christina