I know that I've heard the term "could computing" many times, but I've never fully understood what it means. This post explains what the cloud is exactly, some of its benefits and uses,

In general, the cloud is a network of servers, each with a different function. Servers can be used to run applications, deliver a "service," or store data. The cloud is a metaphor for the internet; cloud computing means storing and accessing data and programs through the internet instead of a local storage area like your computer's hard drive. Local storage is fast and easy, but for those on the local network. With the cloud, cloud computing can be done anywhere, anytime with an online connection. For example, pictures you take on your phone are stored on the phone's internal hard drive; however, when you upload a picture to Instagram or Facebook, you are storing the information on the cloud. Chances are, you encounter and use the cloud all the time. Applications and services like Google Drive, iCloud, Dropbox, even Netflix all use the cloud. Some services work in the cloud and store a synced version of your files online but also sync those files with local storage. When your data is stored "in the cloud," that basically means that it is stored on a remote server owned by a company like Apple or Google. Services that use the cloud pay these companies a membership fee to provide this remote storage area.

Benefits:
Companies used to have to buy all of their own storage and hardware equipment. With the introduction of the cloud, companies only have to pay for what they use. This makes it easy for a company to scale up or down, and they don't have to worry about staying up to date with storage equipment. This creates a company that is more cost effective and efficient. Cloud computing also makes it easy for a company to back up its data; it provides a way out if a company loses all of its local data or its local servers crash. With cloud computing, automatic software updates are also an added benefit, as well as the ability to work from anywhere, anytime. Documents are easily shareable among multiple people, and this means that the environmental impact of paper use can be easily reduced.
Arguments against the cloud:
The intellectual property line for information stored in the cloud can be very blurred: it can be hard to tell what information you own, and what information is owned by the company storing it. A provider could have a strong claim on the data you create in the cloud vs. the data you upload to the cloud. Furthermore, your internet service provider/ telecom company controls your access, and companies can continue to make you pay for doing more and more things because they have control over your data. Also, if your information access is unavailable or offline, you will, in most cases, be unable to work with data stored in the cloud. Although most cloud service providers implement great security standards and certifications, using external service providers always opens the door to security risks. You provide the provider with access to important data, and because that data is stored on the server of someone else, you are giving your trust (and money) to them to protect it.
References:
Content:
http://mashable.com/2013/08/26/what-is-the-cloud/#SZ5hx7iBZkqZ
http://gizmodo.com/what-is-the-cloud-and-where-is-it-1682276210
http://www.pcmag.com/article2/0,2817,2372163,00.asp
Images:
http://www.pcmag.com/article2/0,2817,2372163,00.asp
https://www.cloudbuckit.com/images/blog/56/cloud-computing.png