Building a REST API for Question Answering With Haystack involves creating an application programming interface (API) that allows individual computer systems to interact with each other. A question answering API is a production-ready way to embed semantic search capabilities into a website or mobile app, and can be used for various use cases such as question answering functionality on websites or in mobile apps, generative question answering, FAQ search, and document search. To deploy a Haystack API, one needs to initialize the Haystack infrastructure, add data to the underlying datastore like Elasticsearch, start the Haystack API container, and query the API using HTTP requests. The Haystack API can be exposed on a public IP address by modifying the AWS security group associated with the EC2 instance, and custom API endpoints can be added using FastAPI.