IaaS gives you a server (virtual machine or VM) in the cloud that you have complete control over.
With an Azure VM, you are responsible for managing everything from the operating system on up to the application you are running. This cloud model will feel most like a typical on-premise VM where you use a ‘remote desktop connection’ (RDC) into the server to manage it.
On Azure, these VMs can be scaled up or down either manually or automatically to meet changing performance requirements and to manage costs.
Paas - Means "Platform-as-a-Service"
PaaS solutions appeal to developers who want to spend more time coding, testing, and deploying their applications instead of dealing with hardware-oriented tasks such as managing security patches and operating system updates. It will also save IT admins a large amount of time since they no longer have a VM or operating system to manage and backup.
Azure App Service is a PaaS solution. It is by far, the easiest, cheapest, and fastest method of getting your application to the cloud.
Azure App Service gives you the least amount of configuration control over your application, which is perfect for simple web applications that do not require more complex configurations for scaling and data processing.
You can even include a MySQL or SQL Server database when provisioning an Azure Website through the Azure Management Portal. Once your Website has been configured, simply update your Web.config (or similar) with your new database’s credentials.
Saas - Means "Software-as-a-Service"
Most SaaS applications today are built on a cloud platform due to the low cost of entry – with prices continually falling – and the ability to scale up as your business needs grow.
You can select and configure many applications from the Azure Marketplace such as a fully functional WordPress site with a MySQL database within a few minutes as a SaaS offering. No code, no deployment hassles, and minimal configuration.
SharePoint Online, when you extend its functionality, is generally considered a PaaS solution even though using its built-in functionality only may be considered a SaaS solution. It depends on how you use it.
Office 365 and Exchange Online are SaaS solutions running on Azure that will allow you to replace your on-premise hardware and software without a migration to the cloud.