Azure App Service (used to be called Azure Web Sites) is a "PaaS" (Platform as a Service)
that is based on Windows VMs, at least it was until recently.
Now, there is also a Linux based option, and that is based on containers!
So not only the web apps can run on Linux platform, the users can deploy a (Linux based) containers in a very simple way! Since .NET core can run on Linux (containers) also the whole deployment can be done directly from Visual Studio (maybe even from a Mac).
Azure App Service is the most convenient way to deploy web applications on Azure
and since even a single instance comes with 99.95 SLA it is also most const effective for small to medium loads.
With so much news from Connect 2016 this useful info didn't get much attention.
It is clearer now thanks to this interview with "person in charge" of .NET and ASP.NET.
podcast: Connect Debrief with Scott Hunter @ .NET Rocks! vNext
App Service on Linux now supports Containers and ASP.NET Core | Blog | Microsoft Azure
"App Service provides default Linux containers for versions of Node.js, PHP and ASP.NET Core that make it easy to quickly get up and running on the service. With our new container support, developers can now create customized containers based on the defaults."
Azure App Service improves Node.js and PHP developer experience | Blog | Microsoft Azure
App Service on Linux | Microsoft Docs
How to create a web app with App Service on Linux | Microsoft Docs