"Cloud Computing" is a general term that denotes a service provided by a service provider, typically accessed via the Internet. For example, this web site is hosted by an ISP (Internet Service Provider) and is available via the internet. Cloud also includes services that are hosted by and ISP, that are exclusively available to your company via a Virtual Private Network (VPN) connection. The type of services can vary from full server environments to specific services like a storage archive or a web site.
When discussing cloud computing, it is often helpful to differentiate what service(s) are needed, and whether the hosting environment is private (available only from your corporate network), or public (offering services to external users via the Internet).
The attraction of Cloud Computing is that you only pay for what you use, and you are able to scale up or down based on the current demand.