The easiest way to demystify 'The Cloud' is to compare it to what you already have at home, or in your business. A computer, or many computers, with your information stored and processed on them.
The cloud is exactly the same thing, except the computers with your information are stored in a secure data-centre either in New Zealand or overseas. You are able to access this data over the internet using a secure connection.
Public & Private Cloud
Public cloud, simply put, is using a more subscription like service such as Gmail, or Instagram at the small level, to using a full IT Server solution platform like Microsoft Azure and Amazon Web Services (AWS). The usage varies, but at the end of the day we can say that these services are publicly available for consumption, and in some cases the information for many users is shared in the same place.
Private cloud is quite different. Imagine that we simply took the computer(s) from your home or small business, put them in a datacentre, and allowed you to access them over a secure internet connection. The computers remain yours, but are hosted, and sometimes even managed by a third party service provider. Only your information resides on these computers. Providers can also provision new computing power for you inside the data-centre.
Why is it called Cloud Computing?
Well, the marketing gurus didn't think 'Remote systems accessible online' had much of a ring to it.
To be more honest, it is a happy quirk of IT design evolution where solution designers would use a cloud in their diagrams sometimes instead of a box.
It designs are rather tiresome to look at even for some experts.