Cloud computing is implemented via a 'large' number of computers connected through a network that provide one more services to multiple customers. Common services include computer storage (file storage), software platforms (e.g. Linux or Windows systems) and software frameworks (e.g. database),  In practice, it means individuals and large corporations use external computer resources available via the Internet rather than purchasing and deploying their own systems. It is typically sold by the minute or hour and a customer can use as little or as much as needed at any given time.  It is currently a very 'hot' business model, with Amazon, Google, IBM, ATT and others providing computing services in this fast growing market.
For more information: 

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• Introduction from SearchCloudComputering web site
• Gartner Article
• InfoWorld Article
• A short Goldman Sachs video on recent trends in software, including Open Source and Cloud Computing