In the simplest terms, cloud computing means storing and accessing data and programs over the Internet instead of your computer's hard drive. The cloud is just a metaphor for the Internet. It goes back to the days of flowcharts and presentations that would represent the gigantic server-farm infrastructure of the Internet as nothing but a puffy, white cumulonimbus cloud, accepting connections and doling out information as it floats.
The cloud is also not about having a dedicated network attached storage (NAS) hardware or server in residence. Storing data on a home or office network does not count as utilizing the cloud.