Cloud Computing is basically using the computing resources of other machines that are connected with your computer through some kind of network ( usually the Internet ). It is very useful when you don't have capable enough machine to handle some sort of tasks.
Example: We need to sort a lot of data, we send it to the cloud, who then sorts it and sends it back to us.
Why use it?
Let's say you start a small business and you need a database and some software. For that you'd need a lot of money, time and people. Time is needed to find a place for the equipment, install the equipment, hire the people and get things running. Money is needed for renting a place, buying the equipment, software and paying the people who maintain it. Instead of just getting through with all this, it's cheaper and easier to start off with using cloud computing. You'll get a database that has a storage place and people already maintaining it, a running software, and you won't have to spend any time on installment or hiring.
Using cloud is very mobile, it can be used in any place that has connectivity to the network ( clouds are mostly built on internet ).
Cloud for regular people
Storing data and getting to use it anywhere you go, it's easy and you won't have to use any physical devices. For example, you need to give some files to a friend you're about to visit, but you have no other storing device or you just don't want to use any other devices.
Dropbox and Tonido . You get some free space, but you can always buy more.
Playing games online. If you don't have a powerful enough computer to play the games you like, use cloud gaming. There are also free trials.
Onlive . Some games are free and some offer free trials, but mostly you'd have to buy the game.