'Cloud hosting' can be an ambiguous term. That ambiguity can make shopping for cloud hosting difficult. How are you to know what type of cloud hosting you need when you can hardly define it? Much like their Cirrus and Cumulonimbus brethren in nature, cloud hosting platforms come in a number of shapes and sizes, but also like their puffy floating namesake, once you learn a little bit about how cloud hosting works, it's not hard to figure out what type of cloud you are.
At a very high level, there are only a couple of different configurations a cloud hosting account can take. But the differences between these configurations are very important. When you lease a dedicated server, you get your own machine and your own resources. These resources are devoted entirely to your operations; you can use them to serve web pages, crunch numbers or attempt to calculate the "great question of life, the universe, and everything."
Because the server's entire resources are devoted to you and available to you at will, you can freely deploy these resources within the servers limitations. In a shared hosting account, you share a server with a lot of other people. Because there are many people using the server, your usage is restricted. Chances are, you won't be able to calculate 42, but you will be able to serve up web pages. The difference between shared and dedicated clouds is very similar to the difference between shared hosting and dedicated servers.
Private or Public Cloud?
A public cloud is similar to a shared hosting account because you share one common infrastructure with many different people. A shared cloud generally means that you purchase resources on a utility-computing model.
Wikipedia defines cloud hosting as "Internet-based computing, whereby shared resources, software and information are provided to computers and other devices on-demand, like electricity." In a public or shared cloud, if you need more electricity (server resources) then you can simply use more and you will be charged for more. If you're going be out of town, you can turn the lights off (stop using resources) and you won't be charged at all. Public clouds offer a lot of flexibility in both directions, and are great for businesses that have rapid-fire-like spikes in utilization that are difficult to predict in advance.
However, because public (shared) clouds do share resources, there are a lot of concerns about privacy and security. Also, because they are shared, the end users typically are unable to make certain changes to the server configuration, and work within the bounds of the cloud computing environment.
Private (or dedicated) cloud hosting is very similar to shared (public) cloud hosting on a number of levels. Like a shared cloud, servers on a dedicated cloud can scale upwards and upwards, adding resources as needed on the fly. This means that, like a shared cloud, there is theoretically no limitation to computing power and expansion. As additional resources are required, additional virtual machines can be added. As resource requirements diminish, the virtual machines can be scaled down to take up minimum resources.
However, unlike shared cloud platforms, dedicated clouds offer devoted resources. Like dedicated servers, they are entirely isolated from other sites or users, and thus are devoted 100% to you and to calculating your plans for world domination (or whatever you would do with such awesome computing power!) A dedicated cloud also has one additional advantage, and that is privacy and security. Again, because the cloud is private and isolated, no one else can use it or get into it. Thus, it is appropriate to use a dedicated cloud to handle sensitive information where one may be concerned about security and privacy.