In layman terms, it simply refers to the trend of shifting computing applications and resources to a network of machines (the "cloud"), rather than have them reside in individual, standalone computers.
This has been made possible by the rapidly decreasing cost of computer storage and network bandwidth. A good example of "cloud computing" in practice is
Google Apps. It provides a full suite of applications that businesses or consumers typically use, such as e-mail, chat-conferencing, word-processing tools, presentation tools, spreadsheets and so on.
In the past, a business would have to pay a licence fee to use these tools, and it would also have to host these applications on its own servers, which it also uses to store its corporate information. Now, however, the business can simply make use of Google Apps to set up its own "enterprise" system and, theoretically, not have to care about where its information and applications are being stored, because Google will take care of those details in the background.
So, in a way, "cloud computing" is about outsourcing your applications and data storage to a third party. The obvious advantages are reduced overheads, and distributed computing: You can access those tools and data wherever you can get an Internet connection.
The obvious
disadvantage is the potential loss of privacy and security. How much do you trust the third party, Google for example, to keep your data safe?