The traditional model for business applications are either locally-installed software on individual machines or local client software that accesses server-installed applications. In both instances, the user machine is required to have specific software installed to make the applications run. To run applications completely in the cloud is an old concept with a new name, but it is gaining popularity.
Most cloud applications run inside web browsers. That means the user does not need anything extraordinary installed, and any device with a browser can have access to all the business applications the user needs to work. Browser-based applications have been around for quite a while, but web technology has advanced significantly in the past ten years, greatly increasing the ability of developers to deliver web apps that feel like desktop apps.
The current cloud technology requires the user to work and also save data in the cloud, but this will eventually change, as HTML5 will also give users the option for local storage. This makes it easier for users to pick up right where they left off, even after closing the browser window accidentally.
In the case of public clouds, the application remains in the hands of the provider. The user does not have to worry about maintenance, upgrades, or any backend issues. It also means that the provider has total control and may use proprietary data storage or other methods that might make it difficult to migrate to another system.
In a private cloud, the business has an exclusive server (either remote or local) that hosts its web applications. The provider may still have ultimate control over the applications, but this is not a necessary requirement. Some businesses may prefer this, especially if the provider allows for more customization.