TL;DR: "The Cloud" is just a remote network of computers whose resources you have access to over the internet.
What is "The Cloud"? You've heard about it, everyone's talking about it, but what really is it? To understand what this buzz-word means, we first need a very basic understanding of how the internet works.
It's fair to say most of us are familiar with files on our computers. You can open up Microsoft Word, hit "Save As", and a file is saved somewhere on your hard drive.
But let's say I want to see that Word document (or any other file on your hard drive for that matter). In order to do that, you need to install a special piece of software called a server, which does what it sounds like: it serves files from your computer to others who wish to access them. The network of machines collectively serving files to anybody and everybody makes up what we call the internet. In turn, I use a browser to get at those remote files. So just like that, I can retrieve files from a remote machine.
Now, when we talk about "The Cloud", we are typically talking about one of two things: Cloud Storage, or Cloud Computing Services (which includes the former).
Cloud Storage is the ability for me to not just retrieve files from your computer or server, but also keep my files on it. So in addition to simply grabbing things off, I can now put things on your machine. This saves me valuable hard drive storage space (sometimes referred to as "memory", though it's a misnomer) on my computer! Of course, it comes at a cost to you, so I might pay you a monthly fee to keep my files safe and sound. This is how Google Docs/Drive, Dropbox, Box, iCloud, and SkyDrive/OneDrive operate, though at a much, much larger scale: rather than one computer/server, they might have millions!
Cloud Computing Services (sometimes also known as Software-as-a-Service or SaaS) aren't just limited to providing hard drive space. They might mean a server provides me other services. For example, fairly Wolfram Alpha is a well-known service that you enter in questions, such as "what is 9 * 5 + 23 / 6312?" or "was it raining on 12/22/1993?", and are provided with answers. It works by evaluating what you type in through natural language processing and performing computationally expensive (read: really hard) tasks like searching a huge database of information, and returning to you an answer. All these things are difficult to do on one computer, so they're done on several hundreds of them and returned to you over the internet.
So with all this magic going on, you might be wondering, what are the downsides? Well, for one, you need an internet connection. Wolfram Alpha is rendered useless if it can't communicate with the computing clusters it retrieves answers from, and you can't access those pictures you stored on Google Drive if you don't have an internet connection, or if Google's connection were to fail (something that rarely, rarely happens). Additionally, as we all know, the internet is not instant. It can be slow, so storing and retrieving large files can be something of a hassle. Finally, we must also question the security issues: if your cloud provider has a security issue, it could affect your files' security or your ability to access your files.
That said, the benefits provided are also great. In the future, it's very possible you won't need a super-expensive computer: you'll just need a screen and an internet connection. You'll be able to remotely connect to a server hundreds of times faster than anything you could keep at home, and for a much lower price. Cloud companies can also afford to keep your data backed up, so you won't lose your files like you would if your computer or hard drive failed. In addition, although your individual computer is less enticing of a target to hackers than a huge cloud storage firm, it's also true that the cloud company has a vested interest in keeping your stuff safe, and can invest much more in reinforcing their security than you could.
I hope this helps give you a good idea of what the mystical "Cloud" is. If it did, feel free to share it, or leave a comment below if you have a certain computing term you'd like explained. Congratulations! You now know more than most about what the internet is and how the cloud works.