The cloud is a wildly inaccurate metaphor, a marketing buzzword, and a legitimate approach to web hosting.
Web infrastructure providers have access to vast amounts of computing power — nearly countless stacks of server hardware and storage, connected together to form amorphous computer networks that can act as a giant supercomputers, and which can be then partitioned off into virtual computers of highly variable size and configuration.
This is "the cloud." It's a wildly inaccurate metaphor because there is nothing cloudy about it. It isn't a weightless mist of computer ability — it is stacks and stacks of very real computers, in gigantic air-conditioned rooms, connected together with copper cables and fiberoptics.
The cloud is also a marketing buzzword. Web hosting companies like to use it to cover over the fact that they are just renting their computing power from another provider — often from one you could rent from yourself for less money. Software companies sell software as a service subscriptions on top of "cloud based" infrastructure because it's cheaper for them, but they market this fact as if it seriously makes a difference to you, the end user.
All that being said, cloud computing is a completely legitimate approach to web site and web application hosting.
If you are building a new web application, or a for profit web site, and you need it to be able to scale up to handle large amounts of traffic, the most economical solution is probably to use cloud hosting. Because your site or app is being host on a virtual machine, running on top of a variable pool of computing power, cloud hosting makes it easier to increase or decrease the amount of available power as needed. This can keep your site running no matter what kind of traffic you experience, and can help keep your expenses down by allowing you to only pay for what you use, and only use what you actually need.