Cloud computing refers to network-based computing in which the identity of individual components within a large-scale set of computing infrastructure is given less importance, and emphasis is placed on the actual work done within "the cloud". In other words, rather than depending on specific machines set up to perform a particular task, computing work is carried out by machines (often virtual machines) within a cluster. The identity of those machines is seldom important, as the work could be carried out by any of the machines in the cluster.
The intention behind the use of the "cloud" terminology is to emphasize the effectively anonymous nature of machines in the cluster; the computation takes place "up in the cloud" (insert vague hand-waving gestures here) rather than on specific, identified machines.
I migrated my own hosting to the cloud in February 2010, specifically to AWS.