Public Clouds: An open and publicly accessible cloud environment. Located externally, providers either rent out their resources or make them freely available.

Private Clouds: A custom design cloud computing environment within a protected firewall. Such clouds are usually highly expensive. 

Hybrid Cloud: A mix of public and private clouds. In most cases deployed to boost an organization's existing computing infrastructure.

(Source: Aaron M. Cohen, Types of Clouds, THE FUTURIST, July-August 2009)

Cloud Computing vendors normally specialized in one of the following models.

SaaS- Software as a Service - saleforce.com

IaaS - Infrastructure as a Service - Amazon EC2

PaaS - Platform as a Service - Facebook