Microsoft SharePoint Online, Microsoft SharePoint Services and Azure

This article is about Microsoft SharePoint Online and the Microsoft SharePoint Services which is also sometimes mentioned as SharePoint in the cloud. It is not about the Windows SharePoint Services which have to be installed on a server. So what are these things and what can I do with them? In order to understand Microsoft SharePoint Online and the Microsoft SharePoint Services we first need to understand Application Platforms in general and the Azure Services Platform.