Cloud is where a server, like your email, stores the files that make that server work. It is not actually in your computer. Instead, it is in what the experts call, The Cloud. The Cloud is basically a publicly-accessible giant file system where companies store data that users can access.
It has over 1 Exabyte of data (2013), and to put that into perspective, over 1*10 to the twelfth power, which is 1,000,000,000,000 megabytes of data, which is over 7,812,500,000 times the data of the iPhone 6, which is considered to be a lot. Also, the technology in the iPhone 6 is more advanced than the technology used to send Neil Armstrong to the moon, so imagine the capabilities of Cloud.
What happens on the client side is:
Server side:
Each application has own server.
A central server monitors traffic on all other servers.