The cloud is a semi-generic term for online storage solutions. This can range from something like dropbox that gives file storage to Google's cloud, which is used to sync various data points such as contacts, calendar, even music. Most have a limit on their free amounts, with payment required for a larger limit. In general, if you sign into your gmail account on your Android phone (required to use many of the free services, or to access the Play Store), there is no extra charge.
The cloud is a system whereby you can pull files (photos, music, etc.) from your computer to your cell phone when you're away from home. It does eat data, so if you are in one of those limited data plans, be careful. Oh, one catch, your computer must be on. It's a handy little gadget started by Microsoft. Now Motorola has it's own version, as does Google.