A definition of a cloud's and the related operating systems that I have seen published from VMware is as follows :

"Cloud Operating System
A cloud operating system is a new category of software that is specifically designed to holistically manage large collections of infrastructure – CPUs, storage, networking – as a seamless, flexible and dynamic operating environment.  Analogous to the operating system that manages the complexity of an individual machine, the cloud operating system manages the complexity of a datacenter.  Although alternative approaches may be pursued, VMware believes that virtualization is the key underpinning technology to enable the cloud.

Internal Cloud
An internal cloud is a cloud infrastructure provided within the internal IT environment, which brings the benefits of cloud computing to internal IT service providers.  Gartner refers to an internal cloud as a private cloud. 

External Cloud
An external cloud is a cloud infrastructure that is owned and managed by an external IT service provider, such as a hosting or software-as-a-service (SAAS) provider.  This is often referred to as a public cloud infrastructure as well. 
Virtual Private Cloud
A virtual private cloud is a cloud computing environment that spans internal and external cloud infrastructure, presenting a seamless, managed cloud to the business.  "

Wikipedia quotes cloud computing as :

"Cloud computing is a style of computing in which dynamically scalable and often virtualized resources are provided as a service over the Internet"

The over the internet term is a bit ambiguous and incorrect in my humble opinion.
I prefer something like this :

"Cloud Computing is about dynamic scalable resources, provisioned on demand to meet the needs of an end user"

The best site I have seen for demystifying the FUD and actually educating people on what a cloud is http://www.rackspace.co.uk where the image below has been taken from.