A Private Cloud is simply a network that you are able to run without exposing it to the public internet. Many applications do not need to be accessed by the outside world due to security reasons. A great example is a corporate database. These systems hold proprietary information and are not exposed publically to anyone on the internet. Instead, they live in a private cloud where only specific applications that need access to the database are granted permission to do so. This is enabled by launching applications and databases into a Private Cloud without exposing it to the greater internet. All of the major cloud providers offer mechanisms to customers that allow them to choose whether or not they want their applications in a private or public cloud.
in other words
The equivalent to flying private.