While the term cloud generally considered as an Internet cloud, some large organizations that may also implement private clouds, especially those types of organizations that provide services to themselves, such as a very large organization with divisions or sub-organizations. Rather than these individual divisions managing their own IT resources, the work-loads are shifted to a private cloud.
Basically in a cloud computing system whether its public or private, there is a shift in processing. Local computers no longer have to do all processing with regard to storing data or running applications. The network of computers that make up the cloud takes care of this. The requirements for hardware an software on the client computers diminishes. With cloud computing, the user's computer needs only needs to run a browser with an network/Internet connection to access the "cloud".
The main difference between a public and private cloud is simply whether one or more organizations are sharing resources in the cloud.