The blog you build on WordPress.com will be sufficient for many businesses. As you become more proficient with blogging and WordPress, you may want to create your own self-hosted blog using WordPress.org software which is free.
The software is free, but you have to install it on a hosting service that you select and pay for. There are many hosting choices, and cheap is not better (says the voice of sad experience). Building your own self-hosted WordPress.org website allows you to use any theme and any plugin, and expands other options like sources of video. Amit Agarwal wrote a detailed post on How to Migrate your Blog from WordPress.com to a Personal Domain. I endorse his recommendation that you buy your domain name DIRECTLY from a domain registrar. Don’t buy the “hosting with free domain name” combo. You will spend about $10/year more to register your domain name separately from your host but it will save you countless headaches down the road. It took me many hours of work to recapture my domain name from Sonic.net. Even if you choose to host with Sonic.net, register your domain name through a registrar like GoDaddy.com, NetworkSolutions.com, or Google Apps.
When Amit talks about “renting server space,” it means the same thing as signing up with a hosting service. The WordPress statistics that are built-in to your WordPress.com website are easier to find than Google analytics and may be sufficient. If you don’t plan to put “ads by Google” on your business website, the website you build on WordPress.com may be perfect for right now. You can attach your business domain name to your WordPress.com blog for a few dollars a year using the Upgrade tool. Amit’s blogpost shows you how.