Mod Loaders
Mod Loaders are used to inject modifications into the game.
In simple terms, they handle loading Mods, and provide tools for create such Mods.
Available Loaders and their stories
Neo Forge
NeoForge is a fork of LexForge, created from The Forge Fork.
NeoForge improves on the complexity and rough parts of Forge, along with other improvements, such as using Fabric's Mixin, among other things. We recommend checking out their blog for more details: neoforged.net
NeoForge is available for the latest versions of Minecraft, down to 1.20.1, and has Snapshot builds.
Fabric
Fabric was born from Forge during development of the Mod Loader for Minecraft 1.13, due to slow progress, Forge's bloat, and Lex Manos being a jerk.
Fabric is designed to be quick to update, so downtime between availability when a new version of Minecraft is released is minimal.
Fabric is available for the latest versions of Minecraft, down to 1.14.
Quilt
Quilt was born from Fabric, developed by a very vocal trans activist group, due to the group making demands to Fabric moderation, eventually being unable to get what they wanted.
Quilt is not very interesting and unique, which is why it has a quite small userbase.
It also supports Fabric mods, basically making the loader redundant.
Quilt is available for the latest versions of Minecraft, down to 1.14.4, and has Snapshot builds.
Rift
Rift was born from LexForge during development of the Mod Loader for Minecraft 1.13, due to slow progress, Forge's bloat, and Lex Manos being a jerk.
Rift was designed similarly to Fabric, it was decently lightweight by not providing tools Modders could develop themselves.
Rift was not available for long, partly due to it being criticized by using MCP Mappings where they shouldn't have.
Rift is available for Minecraft 1.13 to 1.13.2.
LiteLoader
LiteLoader is a client-side Mod Loader, and as the name suggests, is lightweight.
Mods developed on LiteLoader cannot have features such as custom Items, Blocks, etc. due to its lack of server-side support.
For that, LiteLoader can be used with LexForge mods.
LiteLoader is available for Minecraft 1.12.2, down to 1.5.2.
Risugami's Mod Loader
Risugami's Mod Loader was developed during Minecraft's beta phase, up to 1.6.2.
This was the first unofficial Mod Loader, before LexForge.
That's really the only notable thing about this Mod Loader. Any other information for this would be nice to have.
Forge, aka LexForge
LexForge was for a long time, the main option for a Minecraft Mod Loader, founded by Lex Manos, and supports versions from 1.1 to 1.21.
The reason it's called LexForge is due to the Forge Fork.
More info here: The Forge Fork
Forge is available for the latest versions of Minecraft, all the way down to 1.1.