This information is up to date as of November 2015. The Azure offer changes a lot, so this information may become quickly out of date.
A few years back, Scott Hanselman wrote a blog post on how you could utilize the "Standard" tier of Azure websites to save money by hosting multiple sites. Well, that was back in 2013 and the azure offer has significantly changed.
Firstly, Azure Websites has now been merged under Azure App Service, along with a few other services. Here is a 5 minute video from channel 9 explaining what exactly is in Azure App Services.
So, how do I get that "Shared" tier multiple websites setup that Scott Hanselman originally blogged about? Well, the Azure App Service pricing details page looks like you can get there with the "Basic" tier, which is cheaper than "Standard":
Azure App Service offer. Screenshot grabbed in November 2015
And how do I actually set this up in my Azure portal?
Confusingly, what is priced as an Azure App Service basically means everything under the "Web + Mobile" under the "new" option in the azure portal:
An App Service plan is the container for your app. The App Service plan settings will determine the location, features, cost and compute resources associated with your app.
More info here.
You get told this information when setting up your app service plan and location (not sure why it defaults to Brazil...)
So, an App Service Plan is basically the billable container for your apps.
So if you want to create a new web app under the same App Service, simply select it when setting up a new Web App.