Overview
The "cloud" term is overused and can refer to several different scenarios. But they all share in common that the networking, software, or website services are being provided at a location not owned by your company. If you're a small business that can't afford onsite IT, or a true server room, or needs redundancy, then cloud services may be right for you.
Here are the three types that I work with most often: virtual private servers, co-location, and cloud services.
Virtual Private Servers (VPS)
For solutions that can run on a single server and don't need redundancy, VPS is a great low-cost solution. The server runs in their environment, there are no licensing costs, and they usually have redundanct internet connections for high availability. There are many providers, GoDaddy being a decent one.
I currently support several VPS solutions, and also use it for my test server
Co-Location / Hosting Facility
For more complex solutions requiring multiple servers, firewalls, and/or switches, hosting facilities provide a rack for all your servers and provide a secure locatoin, cooling, power, and internet. You pay based on how many racks are needed, and how much internet bandwidth. Hardware cost, licensing, and configuration are all handled by your IT. It's preferable to have the facility close to your IT guy; there are lots of them in Phoenix and Las Vegas, so that works well if I am your IT support.
The facilities typically have high-availability internet and power backup; however, all of your equipment is still in one location. A natural disaster or hardware failure can take the system down.
Cloud Services
There are not any hardware or licensing costs with most of these, and they can provide any sort of network, office, or software solution. What makes a cloud solution "true" for me is that it offers scalability and geographic redundancy. For a cost, you can pay to have your system copied in real-time to servers on other continents. If you serve customers around the world, they will get faster results from a system closer to them.
If your traffic/demand varies greatly from one day/hour to the next, rather than buying a server powerful enough to handle the high demand times which is then wasted during low-demand times, with cloud scalability you can only pay for what is needed. Resources will increase/decrease with demand. Most applications have to be changed, or designed, with this scalability in mind, and that is something I can do.
Amazon Services, Google, and Microsoft Azure are the big cloud service providers. I am very familiar with creating solutions to run on Microsoft Azure.