<p>Could anybody explain in plain words how Cloud computing works?
I have read the <a href="http://en.wikipedia.org/wiki/Cloud_computing" rel="noreferrer">Wikipedia article</a>, but still not sure that I understand how cloud actually works.</p>

Aside from the latest marketing term?
Basically all the resources your program needs are held "somewhere" on the internet. You interact with them via a defined service contract; SOAP, REST, POX or whatever and what happens after that is up to the service provider. You don't care about how your information is stored or how the service is provided, just that it is.
If, for example, you wanted to store files, you may choose to use Amazon's S3 cloud system. You connect to the service and upload your files; you don't know or care where the files are stored, only the location of the entry point to that service.
If you have an application then it may also be ran in the cloud, assuming it's suitable. Live Mesh for example is a virtual machine which you can code against and run your software both locally and within the cloud, so your user simply goes to a URI and finds your program, you don't care where it is beyond it being available somewhere on the cloud.