'The Cloud' is a pretty awful example of tech jargon, and it's real meaning is debatable. It does however have vernacular meaning, but it's a meaning that's not well understood outside of techy professionals and hobbyists and somewhat confused by the mixed, awkward messages of different technology companies offering 'Cloud-based' services. Cloud computing, by what can be understood as it's common meaning, is the future of many of the functions our computers perform for us.
On it's simplest level, the Cloud eludes to taking a task that our computers do, and rather than doing it on the machine itself, doing it on a remote server (or servers) connected to the internet.
This might be storing your contacts on a central server so that all of your devices synchronise. It might be storing your documents on a remote server instead of your local hard drive so that you can access them from anywhere with any device. It might be giving your phone a processor-intensive task and your phone sending the data to a server to be processed there and the results sent back. If your computer can do it, then someone else's computer on the internet can do it instead - and for some things this has clear advantages (often simple convenience).
The Cloud services that most of us need to understand fal into two broad categories: The syncing service and the file storage service. You'll find that these are pretty blurry distinctions though. Probably the two most prominent (and competing) services are Apple's iCloud and Google Accounts. Both of these offer the centralised storage of your contacts, calendars and e-mail, making them accessible from your iPhone or Android phone, PC or Mac, or anywhere you can get at a web browser. These services (such as webmail) are really just an advancement of the familiar, automating what could be painful computer-syncing procedures. Most of the time, you just enable syncing and the rest happens without your input.
File storage services are a little different - rather than keeping your documents solely on your hard disk the tendency is to create a directory on your disk which is mirrored or copied to a server on the net, and then back to other devices that you configure. Take for example Dropbox: once installed the Dropbox softwar uploads any files you put into the Dropbox folder on your computer to the web server, and downloads any files that you uploaded by dropping them in on another configured machine. Essentially, using the remote server as the master copy, all of your Dropbox folders on every configured machine are kept identical in terms of content. This is great not only for ease of accessibility (no more thumb drives, disks or e-mails to yourself) but is also a good way of backing up your files in case you lose data on your computer itself.
There are music storage and remote access services, online photo galleries, online word processing and spreadsheet suites; any number of practical computing tasks that can be enhanced by hosting them on the internet. Most of the ones that we think of as 'Cloud' services are personal, for private data and informatio - allowing for access to our data, whatever that may be. When you hear the words 'the Cloud', consider what functionality is actually being offered, and conversely if there's something you're doing that could be better facilitated by internet hosting, there is probably an appropriate 'Cloud' service for that.