In the simplest terms, cloud computing means storing and accessing data and programs over the internet instead of your computer's hard drive. (The PCMag Encyclopedia defines it succinctly as "hardware and software services from a provider on the internet.")
Ultimately, 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 cloud, accepting connections and doling out information as it floats.