Before exploring the advantages of cloud computing, perhaps we should define what it is. In the pre-cloud days, if you needed computing power, you bought an actual, physical computer. But from the customer's point of view, we don't necessarily want a computer: we just want to compute. We would like to be able to buy as much or as little compute resource as we happen to need at a given time, without paying a large fixed cost for a dedicated computer.
Enter virtualization. A single physical server can provide a large number of virtual servers, each of which is (in theory) completely isolated from the others. The hosting provider builds a platform (consisting of many physical servers networked together) which provides, from the customer's point of view, a large intangible cloud of virtual compute resources (hence the term).