What is the Cloud?
The Cloud is a networked group of servers that are available for use by companies or consumers where access is controlled by third party vendors. While these computers are connected to the Internet, they are not necessarily part of the Internet. Think of the Cloud as the factories and warehouses and the Internet as the showroom.
What is Cloud Computing?
In its most basic sense, Cloud Computing is accessing a third party's servers to either store, retrieve, or perform a computing function. You are essentially allowing a company to manage your data and perform computations for you using their servers as opposed to your own computers, or network. (There are exceptions to this when a person or a company sets up their own private cloud using their own servers.)
Whether it is your photos, your music, documents, executables, etc... they are stored on a third party's servers and the computing takes place on those servers. Generally, you will access these servers on your own computer via a cloud portal. Most of the time this is as simple as opening a web page in your browser.
The benefits allow for you to have access to virtually unlimited computing resources. Cloud computing is usually charged to the consumer through a subscription, or through metering your use. Sometimes it is subsidized through advertising and offered to you for free.