Warning: This is a basic, simplified view of modding. A lot of terminology is used loosely, and will not be helpful for developing a mod.

Modding is changing (modifiying) a game to work differently.

Fabric is a mod loader. It takes the mods you've installed and wakes them up when the game is running. Other than that, Fabric doesn't do much that you'll see. Most of its job is behind-the-scenes.

When developers write code, sometimes it does the same thing as someone else's code. Although it wastes time, this happens often. Just like everyone, developers like to use tools that others make instead of doing everything by themselves. Those tools come packaged as mods, too. Without them, things are a whole lot harder for developers¹⁾.

Whenever Minecraft updates, code changes. Mods are written to work with a certain set of code - a certain version of Minecraft. Sometimes, the changes are small enough that mods will work anyway. Most of the time, however, developers have to update their code. It can be a lot of work, making sure code works for many different versions, so developers usually choose a few versions and stick to those.

Mod loaders do mostly the same thing, but they do it in different ways. Unless the developer of the mod has put special time and effort into making it compatible with other types of loaders, it will only work for one or sometimes two. Plugins are also very different from mods, so it's rare to see things that have both plugin and mod versions.