A Private Cloud is implemented using a dedicated data center infrastructure of hardware and software that is used privately by an organization. The data center can be on-premises or off-premises. It is not shared with another organization. If the data center is shared, that is a Virtual Private Cloud.
The Cloud Computing Stack in a Private Cloud is dedicated to the organization. If the data center is shared but not the Cloud Computing Stack, that is a Virtual Private Cloud. If both the data center and the Cloud Computing Stack are shared, that is a Public Cloud.
A Private Cloud may participate in a Hybrid Cloud.
The following figure uses Customer Relationship Management (CRM) Software as a Service (SaaS) to illustrate a Private Cloud.
More on the general topic: Types of Clouds in Cloud Computing