Organisations connecting to the Internet in the early 1990s had to sign a usage agreement directly with NSFnet to gain access to large parts of the Public Internet, regardless of what Internet Service Provider they purchased Internet access from.
On April 30, 1995, the NSFNET Backbone Service was successfully transitioned to a new architecture, where traffic is exchanged at interconnection points called NAPs (Network Access Points.)
NSFNet is unrelated to NFSNet, a distributed computing effort to factor large numbers. Also NSFNet has nothing to do with Network File System (NFS), a TCP/IP file sharing protocol like SMB.