Cloud Computing has many aspects. One way to look at cloud is as a storage platform. Companies have server farms with allow you to store files on their sites. Think Google docs, box.com etc. Another part is the cloud systems which provide processing and software implementation online. They are called Infrastructure as a service IAAS and Software As a Service SAAS. Amazon provides service called Amazon Cloud Service where you can rent processing power. Main factor for this deployment is that you don't have to buy any hardware. Also after one project ends you can use the same service for another service. In SaaS companies deploy their systems online so employees can access things online.
All this is possible because of virtualization and increase in internet speeds. You should look into emc systems and VMware if you are interested in cloud systems.
Hope this is a simple explanation.