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SaaS (Software as a Service): This is the most popular form of cloud services. The service provider offers a software to support the service on offer. The software is built by the service provider while the end users can configure it to suit their needs. The clients (end users) however, cannot change or modify the software.
Mozy is an example of SaaS. It is basically a backup service that offers a software to help people back up their data. Thus, you can use the service without actually having to code or buy the software. You just have to pay a monthly or annual fee to use the service.
PaaS (Platform as a Service): Offers a platform to clients for different purposes. For example, the Windows Azure offers a platform to developers to build, test, and host applications that can be accessed by the end users. The end users may or may not know that the application is hosted on the cloud. As mentioned earlier, the storage space for user data may be increased or decreased per the requirement of the applications. As with the SaaS, you do not need to build the platform. You just pay a nominal fee for using the service.
IaaS (Infrastructure as a Service): Offers infrastructure on demand. The infrastructure can be anything from storage servers to applications to operating systems. Office 365 offers a combination of these infrastructure and falls under this category. With Office 365, you can get plenty of applications along with storage space. Buying infrastructure or renting it out in traditional models can be very expensive. When you opt for IaaS, you save a lot on expenses, space, and personnel required to set up and maintain the infrastructure. The cloud service provider takes care of setting up and maintaining the infrastructure. You just pay a fee to use it per your requirements.
As you can see, there are numerous advantages of cloud computing, the most basic ones being remote accessibility, lower costs, and quick re-provisions.
Image credit: http://cloudtimes.org/the-cloud-computing-revelations-of-2010/