Cloud computing is a new trend, high on the hype-cycle right now, where you can "plug in" to computing resources almost like you plug in to the electric grid.
In traditional network diagrams, the Internet (or indeed a private network) was represented by the picture of a cloud. This was meant to represent the fact that the internal nature of the connections in the network did not matter for the purposes of this particular diagram. A cloud is opaque, and its inner workings are unknown, but it does the job of connecting the things we care about, those computers that need the connectivity.
So initially, a cloud was a network, often provided by someone else, that connected computers but whose innards were, happily for all, a black box. The network provider could change the routing, the equipment or even the technology inside the cloud, and no-one using the network would even know. The connectivity would continue to work, and the computers would continue to talk to one another.
Now the cloud analogy is being stretched further, to include cloud computing, where computing resources are made available in the same opaque fashion, for customers to use as they need it, without worrying or knowing about the innards. Just as I can increase my use of the network cloud "as much as I want" and it is the vendor's job to make the network appear infinitely scalable, so it goes with cloud computing. If my website gets a massive spike in usage, my cloud computing vendor just provides more computing resources for me "on demand" - all I have to do is pay for the extra usage at that time.
With a nod to Greg Gretsch who brought this to my attention, and James Hamilton who wrote it up, you can get lots more information about this trend from UC Berkeley's "Above the Clouds" paper, presentation and video.