You will almost certainly have heard companies telling you to move all your data to 'the cloud' and how 'the cloud' is changing everything, but what IS 'the cloud'?
In basic terms, the answer is very simple. You can substitute 'the Cloud' with the phrase 'using the internet to send and receive data from other people's computers'. This is opposed to storing data locally - either directly on your machine / device, or on a server or storage device that is local to your network.
The obvious advantage of storing data in the cloud is that, depending on the service, there's an almost unlimited amount of storage available to you at a fraction of the cost that the same amount of space would cost locally, and expanding the amount of space available can be done at a click of a button rather than having to source, purchase and setup new hardware. Services like Dropbox, OwnCloud, Apple's iCloud and Microsoft's OneDrive fall into this category.
Conversely, the downside is that you no longer have absolute control of the data that you have uploaded. All the while your data is stored on your network, short of someone breaking in and stealing the physical devices that your data is stored on, you know exactly where your data is and (as long as your network is secure) who can access it. As soon as your data is uploaded to the internet, your data is only as secure as the service you have put your trust in has made it. Even then, it is often much more vulnerable to attack through simple attacks like guessing passwords than it would be locally. Therefore, one needs to weigh up the potential cost of the data falling into the wrong hands before deciding whether cost and convenience is more important than security on a case by case basis.
As well as data storage, the cloud can also be used to run services. Again, this means that rather than an app running on your personal machine / device, you're connecting to that app over the internet and running it from someone else's computer (via your web browser usually). Services like Google Apps and Office 365 fall into this category (although the latter is more of a hybrid, as it also installs versions of the Office Suite on your local device as well).