Recommended way to host a WebApi in Azure
For hosting a simple web API (that you can scale according to usage, etc.) you'll want to use Websites. Assuming you're not looking for more complex / heavy-weight features (network configuration, more complex architectures e.g. offloading background processing different instances via queueing mechanisms, RDP into the host machine, etc.), then Websites are becoming the de-facto way to host websites on Azure.
The following page from the Azure documentation will give you a full feature comparison between the two:http://azure.microsoft.com/en-us/documentation/articles/choose-web-site-cloud-service-vm/, but in short, if you simply have a web API project in VS that you want to host in Azure without worrying about the underlying infrastructure, then use Websites.
If you only want to host a Headless (No Web Interface) Web API, I recommend you using Azure Web Service - Web Role.
For Worker Role, it is like a console application that you want to use it to process background task. Normally, we use it to process from Message Queue (Azure Service Bus - Queue).
Azure create a Optimize VM to run those two type of Roles with no unnecessary junks. So you will get the most of it.
However, I still suggest you to read more detail document from azure website to see what environment which is best fit for your long term plan.
You need to login in Azure (https://manage.windowsazure.com) -> Web Sites -> Create New
After that you will find the Name of the Web Site Under the Web Sites.
Click Web Sites -> Select New Created Web Site -> Go TO -> Deployement
In Deployement -> Find -> Integrate source control
Select Appropriate Option i.e GitHub or Dropbox etc..
Using Dropbox => Now Publish your WebApi Project and Paste that Data into Dropbox Folder With the same name of your New Created Web Site in Azure
After Upload Go To -> Azure -> Select Web Site -> Deployement -> Sync.
It will take all the data from Dropbox and You can run your WebApi Project From Azure