Cloud computing in simplest terms refers to storing and accessing data and programs over the internet instead of from you local hard drive. The Cloud is really just a metaphor for the internet. It goes back to the days when server farms etc. were depicted as a cloud on presentations.
In personal computing a good example of Cloud Technology is Google Drive, or Dropbox, also Office 365 from Microsoft, another good example would be online banking. These are all storage or office solutions not running on your own hard drive but rather running on servers somewhere on the internet (Cloud).
In business circles good examples of Cloud technology would be Software as a service "SaaS" type applications that the business subscribe to, like Salesforce.com.
Cloud computing also changes the way people and business can pay for the use of a service or program. before Cloud Office programs and other applications had to be bought out right, which meant a big capital outlay for the user. The Cloud on the other hand allows users to pay monthly and only pay for the amount of users actually using the application or service.