Wikipedia defines "cloud computing" as a "paradigm of computing in which dynamically scalable and often virtualized resources are provided as a service over the Internet." For me, it is just a way of computing (or doing your work with your computer) with your data or your applications--or both--on the Internet.
To concretize, look at Google Docs, or Zoho, or ThinkFree. Also, previously, when one conceives of a website, you use either your Notepad or Microsoft Frontpage. Now, you can do so with Google Sites, which reduces your need to learn HTML or similar languages. Further, Google Sites allows collaboration, multiple types of access, and easy addition of content. Imagine if you will have to write all these in code, and you are not a computer science graduate.
Why "cloud compute?" As Eric Knorr and Galen Gruman say, it "comes into focus only when you think about what IT always needs: a way to increase capacity or add capabilities on the fly without investing in new infrastructure, training new personnel, or licensing new software."
The previous paragraph summarizes the benefits. What are the costs? As I see it, it is minimal. Access to the Internet and necessary software for connecting to the internet (which can be free, considering open source solutions), which are all being used anyway.
I will stop here. But I hope this short post stimulates your creativity--focusing on your own interest while expanding your capability at minimal if no cost.