There are several types of cloud computing deployment scenarios.
The National Institute of Standards and Technology (NIST) is emerging as the preferred provider of the de facto definition of cloud computing and the distribution models, seen here with some Esri examples.
The public cloud is the most commonly referenced regarding the topic of cloud computing, where the infrastructure and applications are owned by the organization selling cloud services.
Since many traditional vendors and users are not quite ready to jump into public cloud computing or are restricted from doing so, the cloud service tiers are replicated within a private cloud environment, behind the firewall, and maintained within the parameters of the host organization.
Many believe that the sweet spot for cost optimization in an organization will rely on a delicate balance of public, or community, and private clouds. However, since this hybrid cloud solution is commonly bound together by proprietary technology, it will only be embraced by enterprise computing in the future as standards are developed.