Minecraft Forge is an unofficial API for extending Minecraft. It is built upon the Minecraft Coder Pack (MCP) and is usually updated to the latest version upon release. It is built upon java and has its own initialization system.
Minecraft Forge is an application library for minecraft.
This library is used to compile and play modifications (mods) to Minecraft.
Minecraft Forge works by doing live ASM modifications to the Minecraft classes and injecting its own event busses into the Minecraft classes without permanently changing the original Minecraft classes.
There are different aspects to Minecraft forge. There is the Forge client/server and Forge Mod Loader (FML) which allows players to load the mods into their Minecraft instance. This is used by the players to play and use the mods. When the Minecraft instance starts, Forge injects itself into the Minecraft classes and then starts up an FML instance which checks the "mods folder" for mods and injects those into the proper places.
There is also the development instance of Minecraft Forge which comes as a standalone library/sources or a prepackaged eclipse and intellij gradle setup where you have very little setup to do to start modding Minecraft with Forge. It is the foundation of the majority of current client/server-side mods of Minecraft.
Minecraft Forge builds can be downloaded at the Minecraft Forge Downloads page.
See the Forge Forum or the Forge Documentation for more information on MinecraftForge.