The are many version of the history of the term "cloud" as it relates to computing. In my university days, the "cloud" was called the "internet". The "Cloud" is basically another term for a network of machines. I think the best way to explain this is with the graphic below. In it, all machines and devices that do certain functions, outside of the devices we know and care about, are collected into a cloud shape - usually the big fluffy thing in the middle.
Traditionally, computing has been done with applications installed on a local machine and in more corporate situations, these applications may talk to, and store data on a centralised server inside the corporate network. Cloud computing throws all of that out the window. As shown in the graphic, the only device that people actually use is the screen, the interface. The entire application, storage, processing and communication between interconnected services, is handled by some other device, somewhere else. And that really is the basis behind the "cloud". It's data being processed and loaded for you, by a machine that you don't necessarily know where it is, what it is, or even if it's doing anything else at the same time.