In the simplest terms, cloud computing means storing and accessing data and programs over the Internet instead of your computer's hard drive. The cloud is just a metaphor for the Internet. It goes back to the days of flowcharts and presentations that would represent the gigantic server-farm infrastructure of the Internet as nothing but a puffy, white cumulonimbus cloud, accepting connections and doling out information as it floats.
What cloud computing is not about is your hard drive. When you store data on--or run programs from the hard drive, that's called local storage and computing. Everything you need is physically close to you, which means accessing your data is fast and easy (for that one computer, or others on the local network). Working off your hard drive is how the computer industry functioned for decades and some argue it's still superior to cloud computing, for reasons I'll explain shortly.
The cloud is also not about having a dedicated hardware server in residence. Storing data on a home or office network does not count as utilizing the cloud.
For it to be considered "cloud computing," you need to access your data or your programs over the Internet, or at the very least, have that data synchronized with other information over the Net. In a big business, you may know all there is to know about what's on the other side of the connection; as an individual user, you may never have any idea what kind of massive data-processing is happening on the other end. The end result is the same: with an online connection, cloud computing can be done anywhere, anytime.
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