"The Cloud" is a common term tossed about in todays IT world. What it really means is that all or part of your IT data and/or programs are stored and maintained on a site that you access through the internet.
An example would be if you use Microsoft's Office 365, you e-mail and possibly some data (in SharePoint) is saved on servers that you access through the internet. You could say that your mail and data are stored in the cloud.
There are 3 broad categories for Cloud Usage:
1) e-mail; this is probably the most common as uses of public e-mail services (yahoo, hotmail, gmail, aol, etc) all store the data in the "Cloud" and you access it through the internet from anywhere.
2) Data; the next level of cloud usage is if you store data in the cloud. There are a number of options here, dropbox, dedicated storage from web hosting providers, Microsoft SharePoint, IBM's SmartCloud, etc.
3) Data and Programs; In this model you actually have servers, storage, and desktops (with the programs) in the cloud.