The Nym API provides multiple services to the Nym network, and is designed to be run alongside Nyxd validators. From a logical perspective, there are multiple applications, but they are bundled together for ease of deployment.
The Nym directory is contained in the mixnet smart contract in the Nyx blockchain. The blockchain holds the canonical directory information about nodes, stake, locations etc. The Nym API caches this information periodically to make queries faster and more scalable.
The Nym API periodically sends test packets through the entire of the Nym mixnet to test node liveness and quality of service.
Coconut [paper] is a distributed cryptographic signing scheme providing a high degree of privacy for its users. You can find an overview of how to use it in the Coconut section of the Nym documentation.
A simple explanation is also available in blog form.
This project was partially funded through the NGI0 PET Fund, a fund established by NL.net with financial support from the European Commission's NGI programme, under the aegis of DG Communications Networks, Content and Technology under grant agreement No 825310.