"Cloud computing," while it has very nearly achieved meaningless buzzword status, is an attempt by the business and marketing types to get their heads around what is a very real evolutionary transformation occurring in IT. The drivers are the drying up of CapEx budgets, the need to reduce service delivery time, and the requirement to purchase and pay for only what an organization needs to fulfill their business requirements.
Capital expenditures are coming under increased scrutiny, and are under constant budgetary pressure. "Cloud" based, on demand services allow IT service procurers to shift from a CapEx based model of owning the infrastructure to an OpEx model. Operational expenditures are typically larger chunks of budget than capital purchases, and moving IT services there can allow them to get "lost in the noise". Less red tape, less stringent approval processes, etc.
Time to deliver service is a labor cost, and if a procurer can shift that operational expense from internal overhead required to deploy an IT architecture to acquiring those same services from a cloud provider, it's perceived as a big win. The provider gets to deal with the headaches of capacity management, infrastructure design and integration, and delivering IT resources. The purchaser gets the luxury of simply specifying how much they want, for how long, and letting the provider leverage its economies of scale and automated processes to deliver the resources within the terms of the provider's SLA. The tradeoff is that the consumer of cloud services loses the ability to specify the platform and all its parameters in exchange for rapid delivery of a standardized service.
IT organizations are also under increased pressure to abandon the concept of designing and purchasing for peak capacity. Cloud providers are specifically addressing these needs by allowing their customers to pay only for what they use, not the spare capacity. Since the "cloud" capacity is shared, reused, and managed by the provider, the customer is afforded the ability to scale their environment dynamically to meet the needs of the business and its budget.
Now, how this ties into Web Services is important. Web Services, for a long time, was a solution in search of a well-defined problem. Now, with the "cloud" becoming a workable construct, Web Services come to the forefront as the way that stateless platforms can interact without intimate knowledge of the underlying infrastructure. Web Services will become more and more important as IT services are increasingly abstracted away from the hardware and OS platform. As I've worked for the past two years as a design architect for an infrastructure-as-a-service type platform, I can say with some authority that they're are an integral part of how we're going to need to deal with virtualized environments and stateless service contexts as they become pervasive elements of IT solutions.