With the move towards independent event driven service architecture, in other words Microservices, Microsoft announced Azure Functions at the BUILD2016 conference. The announcement is described in TheNewStack post Microsoft Jumps into Serverless Computing with Azure Functions. The article is worth a read as it describes the offering, offers a comparison with the competition (re., AWS CLI and Google Cloud Functions) and notes the different pricing models available.
For a more comprehensive guide to Azure Functions, Nir Mashkowski provided an excellent post: Introducing Azure Functions. For more detail the Azure Functions documentation contains a clear overview, videos and material for how to to get started.
So, how do Azure Functions fit into Azure when viewed against Api Apps and Logic Apps? Though similar, the intended purpose is different. Logic Apps are aimed more at automating business process by integrated and orchestrating cloud and on-premises resources where Azure Functions are targeted more for building event driven microservices. One post described the difference succinctly by:
Azure Functions is code being triggered by an event.
Logic Apps is a workflow triggered by an event.
Pricing is also different where Functions Pricing is about usage so can be thought of as a serverless model.