Business Central is Microsoft’s ERP offering for small and medium sized business. It is possible to run Business Central both on-premise on your own servers or run it as a cloud based service. There is no doubt that Microsoft is pushing the product in the cloud direction so new customers should really have solid arguments for not running Business Central in the cloud.
Business Central is not sold as a self-service so you will need to purchase your Business Central license through a Microsoft partner. You can however go too http://trials.dynamics.com for a 30 day free trial. Your trial subscription can later be converted into a paid subscription.