ForgedAPI Fork Mod is a dependency, so where it goes depends on the mod that requires it. The simplest rule: put it everywhere the requiring mod is installed.
Singleplayer
1. Open your Forge instance's mods folder.
2. Place ForgedAPI Fork Mod next to the ported mod that needs it.
3. Launch the game. If Forge no longer reports a missing dependency, the setup is working.
Singleplayer runs both a client and an internal server, so one copy covers both sides.
Multiplayer
1. Client: Several of its modules deal with visuals, including rendering, render layers, custom models and the Renderer API with Indigo. Players need it on their own clients whenever a ported mod they run relies on it.
2. Server: It also provides base systems and lifecycle events. If the requiring mod is installed on the server, put ForgedAPI Fork Mod in the server's mods folder as well.
3. Match both sides. If the ported mod has to be on both client and server, this library should be on both too. A mismatch can block players from joining or cause errors on connect.
Quick check
If a player or the server fails to start with a message naming a missing mod, compare that side's mods folder with the other side's. Add the library to whichever side is missing it.