The common part is the "as a service", which means that whatever you're buying, someone else is contracting to maintain it so you don't have to. With *aaS it is common for the vendor to make certain guarantees regarding uptime, how long it'll take them to address outages or issues, and so on. This is called a "service level agreement" or SLA.

Software as a service: I sell you a license to use an instance of my application (say, an issue tracker or webmail). You log in and set things up for your organization. I own the servers. If you have a problem that you can't fix through whatever admin UI you have access to, you have to go through my support team.

Platform as a service: I sell you a license to build and deploy applications that work in my cloud. You write some code that hooks into my APIs and set up some of the environment in a controlled manner, like spinning up a database server with Amazon RDS. I own the cloud and everything you're talking to. If something goes wrong on the hardware end or if you blow up your software development community site by triggering a deploy cascade (😉) you have to go through my support team.

Infrastructure as a service: I sell you a license to use my virtualization framework. You log in and spin up virtual servers, volumes, and networks, and administer them how you like. I own the host machines. If there's a network outage or a meteor hits the data center, fixing it is on me, but if you wipe your own database I don't care. Also, I'm Google or Amazon because you have to be huge for this to make sense as a business model.