There are a few other "clouds" which have nothing to do with this definition. Personal Cloud is one in which basically your data in all your personal devices is aggregated and made available from any device.
Some uses cases are personal storage in the cloud, although that's hardly the only use. Web servers can take advantage of the cloud to become highly available and geographically sensitive. Clouds can be used to rent computing for a one time intensive computing event.
But the general idea is, pay only for what you use, and repurpose what you don't for other tasks.
The main problems with cloud involve bandwidth, industry and country regulations, and portability. Also most applications need to be "modernized" to fully take advantage of the cloud fabric. Just taking your SaaS solution and putting it in the cloud doesn't mean you are a cloud solution, contrary to what some may think.
An interesting resource if you want to learn more about what it takes to create a large scale cloud solution is the Netflix technical blog. They are one of the few customers that survived the Amazon outage since they undestood what they were doing while designing the solution.
