There are a few uses for it, but for the average home user, it's nothing really we need.
Basically, it's using small slices of big powerful computers to do certain things for you on their pc's instead of yours, or storing entire programs, data, etc to be access from anywhere.
Example, you are a physicist, and need to compute some big math numbers. Your workstation PC can't handle it, you sign up for a cloud computing service and something that might take your PC 3 weeks to compute, sending it to the cloud, you can have your answer in an hour because of the mass parallelization of the cloud computing ability. Some charge by time, or computation cycles, amount of data send and received, etc.
Example, you run a small business with workers that use their own equipment and are on the road or job site a lot. They need to access certain spreadsheets or documents or even software when they are away, access from home, etc. Cloud computing can be setup to let users access from any web browser documents, or even software running in the cloud, so they have access anywhere and can all run the same version of a software. You can do this with your own server in your business, but that might require lots of bandwidth which can be pricey, maintenance of the server, security on the server, etc. With cloud computing, they handle all that, and with massive space and computational power, they can provide as much as you need on demand, where your server might start to struggle if everyone connects at once. With cloud power, your share can dynamically use more power and the end user will never get any lag, connection problems, etc. Users can connect with windows, linux, macs, ios, android, whatever they are running because it runs in a web browser. Microsoft offers this with Office365, access your whole office suite and documents via any web browser. Need to update a spreadsheet from an internet cafe, 2 minutes, done.
The most common use for home users is just cloud storage, like google drive, or mircosoft sky drive. You can copy your files to the cloud and access them from everywhere.