The usage of the term "cloud computing" is interesting: it's a widespread buzz word, yet its true meaning is rather nebulous for the average person. As a non-technical business student who initially had trouble grasping the meaning of cloud computing, I'll attempt to explain it in layman's terms.
Back in the old days, if you wanted to create a web application and deploy it to the web, you would need to purchase a hardware server. A server is necessary for running your web pages on the world wide web, so that when someone accesses your site from their computer using a browser (the client), a request can be sent to your server and return the web page back to the user's client. However, purchasing these servers were expensive and inefficient, because if, for example, you only needed the computing power provided by one and a half servers, you would still need to purchase two servers. These costs were prohibitively expensive for most people who now have access to cheap computing power via cloud services.
Cloud computing removes the need to purchase physical servers by providing these services over the web. Now I just need to create an account on Amazon Web Services and set up a virtual server (EC2) and data storage (S3) all on my laptop via the internet. Pricing is usage-based, meaning that I pay for exactly the amount of computing power and data storage I use. Companies that provide cloud computing services (Amazon, Microsoft, Google) purchase server farms themselves and then make money by charging users for access via the internet for computing power and data storage on their servers. This is a significant innovation because it's exponentially cheaper and more efficient for anyone to run a web application. Thus, barriers to entry in the internet space have substantially diminished.
Infrastructure as a Service (IaaS), Platform as a Service (PaaS), and Software as a Service (SaaS) are just fancy classification terms that describe how low-level these services are, or in other words how much granular control you have over your server and data storage systems and application. An IaaS like Amazon Web Services lets you control almost everything from the ground up, such as load balancing (I don't know much here) etc. A PaaS, like Heroku or Google App Engine, lets you develop your own web application and deploy it to their servers but already has the specific server and data storage settings implemented for you. This is a nice option for those who want to quickly deploy to the web and have little to no experience in systems administration. Lastly, SaaS is pretty much everything else. SalesForce is an example of a SaaS for businesses, while web apps like Quora or Twitter could certainly be classified as SaaS for consumers. SaaS is essentially access to a web application that lets you carry out certain functions without having to code it up yourself. I think this is where lots of the confusion surrounding whether cloud computing is truly innovative stems from. No, SaaS would generally not be considered an innovation in the last decade; it's been around longer than that. However, it's not cloud computing as a broad term, but rather specific subsections of cloud computing, particularly PaaS and IaaS, that have revolutionized the online business landscape in the past decade.
