Putting things "in the cloud" has been a bit of an annoying buzz word for a few years, because it doesn't really, in some ways, mean anything other than what the web has always meant. Storing things on servers which are connected 24/7 to the Internet so you can access them from anywhere.
However, "cloud computing" brings with it a little more than just server storage space... it brings scale. And, just as importantly, the ability to scale back. It brings flexibility.
The cloud computing service provider we generally use is AWS - Amazon Web Services. You might just know Amazon as the huge online shop that can get you that thing you forgot you need for tomorrow in time. But for us, Amazon are a beast of a provider of everything and anything you could dream of for making huge websites and web services. They have literally hundreds of different products / services which you can buy to help run your website. I did a training day with them last year and heard how one of their new services was already compatible with 120 or so of their other services and would be compatible with more soon - the amount of different things you can buy from them just goes to show how much engineering is involved with keeping big websites online and secure and operational now a days.
Anywho, I digress slightly. Cloud computing brings with it the flexiblity of just getting what you need, when you need it. For example, if your website traffic always surges at 7pm on week days, you can get more server power at those times automatically by things scaling up when you need them to, and scaling back down again when you don't. That's called autoscaling and is something we do for larger websites we run. Meanwhie medium sized websites just ask us to manually scale things up for a week or so when they're running a campaign and so are expecting a lot of traffic. This way you only pay for the extra resource when you know you need it, rather than having an over spec'd server all of the time.
Something to be wary of with cloud computing though is that it can be very hard to get a handle on what your costs will be before hand. Some things have a monthly fee but generally, as the costs are so granular and almost down to each person and what they do whilst they're on the site, giving an accurate estimate before hand is really tricky. We've got clients on AWS who pay £20pm and clients who pay thousands and thousands (and thousands!) a month. Amazon have an "easy price calculator" which is about the most complicated thing you've ever seen in your life. So really you just need to chat it over with your developer and ask them if they've worked on similar projects and ask them to help you understand the calculator for a guide.
The ability to scale up and down your processing power as and when you need it lets businesses have everything they need without overpaying at quiet times. Data Barracks have put together this great animation with Stephen Fry to explain cloud computing more, and draw parallels with it and essential utilities such as water or electricity, which feels very timely when we've literally been saying in our house lately that having an Internet connection now a days is as essential as electricity.