The following is a very well-known classification of the different type of services offered by Microsoft for on-premises and the cloud. Microsoft Azure provides a full spectrum of services starting with Infrastructure as a Service (IaaS), followed by Platform as a Service (PaaS) and Software as a Service (SaaS) models such as Dynamics CRM, Office 365, and so on:

With IaaS, customers can deploy a virtual machine of their choice without the complexity of buying and managing their own hardware/software; they can quickly scale up and down with demand and pay only for what they use. This will reduce operational expenses of hosting and managing infrastructure.
Azure provides a vast range of PaaS services such as web apps, logic apps, media apps, API apps, mobile apps, Azure Stream Analytics, Azure HDInsight, and Azure Redis...