Fidonet is a non-profit international network of BBSes (Bulletin Board Systems). Services include NetMail (private email), EchoMail (forums/conferences, similar to newsgroups) and file distribution/request system (similar to FTP).
Before the Internet really took off, Fidonet was quite large (over 50,000 nodes) largely run by volunteer "sysops" (system operators) from their homes or offices. It was, in its heyday, the worlds largest distribution channel for freeware and shareware. Now it is less than 30,000 nodes and the number of users per node has dropped significantly.
In theory, Fidonet is a part of the Internet, since the Internet is a "network of networks" of which Fidonet is one such network. In practical terms, however, Fidonet does not support TCP/IP so gateway software is used to facilitate passing of messages between the Internet and Fidonet. Quite a bit of translation has to take place to pass messages between Fidonet and the internet.
Today, Fidonet only survives by a combination of inertia and the efforts of a few sysops who still consider it important enough to carry out the administrative functions of keeping it alive. Some Echoes (Japan Link, for example) have been gated to the Internet via list servers or newsgroups but in the long run, these will be all that survive.
Fidonet echos have ago stopped reaching Japan. My own BBS (Shizuoka Echo) exists today only as a web-based forum.