What is an Azure Queue?
Azure Queues allow you to store a large number of lightweight messages leveraging Azure Storage. Typical use cases for queues include buffering a set of data or actions for processing. And since the Queues are part of Azure Storage they are super cheap. As of writing this, each queue message can be 64 KB in size and a queue can hold millions of messages (up to the capacity of the storage account).
Scott Hanselman has a post showing how an Azure Queue can be used along with Azure WebJobs to offload processing of images. This is a great example as it shows how a real application might accept images from a user in real-time, then use a queue as the hand-off for another service to actually process the images.
As another example, I have used Queues as a temporary staging ground for incoming IoT data from my connected SmartHome where a WebJob then took care of properly sorting the data and moving it to the destination Azure Table Storage location. This allowed me to take advantage of the massively scalable architecture of Azure Queues instead of having my logic running on a Web App where I would have had to take care of scalability.