A REST API is an architectural style for an application program interface (API) that uses HTTP requests to access and use data. It is a way of accessing web services in a simple and flexible way without having any processing. REST technology is generally preferred to the more robust Simple Object Access Protocol (SOAP) technology because REST uses less bandwidth, is simple and flexible, making it more suitable for internet usage.

To implement a REST API in C, there are several libraries and frameworks available, such as:

• Kore: A web application framework for building scalable web APIs in C.
• Casablanca: A C++ REST API framework that can be used in a production environment.
• Restinio: A header-only C++14 library that provides a set of tools for building RESTful HTTP servers.
• Oat++: A C++ web framework that supports URL routing with URL-parameters mapping and Swagger-UI endpoint annotations.
• Crow: A C++ micro web framework that can be used to build RESTful web services.

To call a REST API using C programming that can be flashed into an Arduino board, you can use plain C code that can be burnt into the device. The code should include the necessary libraries and functions to send HTTP requests and receive responses. The code should also include the JSON body that is set up in the format that the API expects, along with a variable to point to the service method.