Homebrew came to the gaming lexicon early on from our friends in the SCA who were brewers (as it was told to me).
In short, you take the basic components provided in a brew recipe and add your own touches by adding/ deleting ingredients or altering the proportions slightly. So a brewer can either do massbrew, recipe-it, or homebrew.
Now extend this to gaming.
Massbrews are the published game settings (or game rules) with no deviations.
Recipe things are combining the raw elements presented and generating something pretty much like everyone else. This is mostly used for game settings that follow the ethos of that comes with the game. Every game has a default world that most GMs modify ever so slightly and run it.
This is your average D20/ D&D world, where the details might change some, but basically if you knew D20/D&D, you can just insert yourself in the game.
Homebrew is a do it yourself. In the beginning, you would take an existing world (like the DnD standard) or game (AD&D was the normal system of choice at the time) and change things around enough to make it your own, your own homebrewed world or game. That was the back in the day. It has been extended now.
The phrase has been extended to those who are making their own unique games or world. This is the next step from the recipe it. Things are significantly different and if you (as a player) don’t know what is really going on you are going to be lost.
We do a great deal of homebrew around here. We do the unique and the unexpected. We do not stand on the expected, we take it and find our own new twist. Our is independent work, making an independent world or game.
Can I just say I have a total loathing for the term Homebrew. I feel it cheapens the efforts of people who go to the effort to make their own game materials. It allows you to disregard it with distain by a simple “oh it is some homebrew”. After all, it is not OFFICIAL or PUBLISHED material, so it must be crap. Yet, we all know some of the best game worlds and systems we have encountered were made by gamers we have met, not publishers. In fact just because they are published, does not mean it is good… it just means they have more time and money invested in it than you do. (Though if it is on the 3rd or 4th printing, there must be something to it.)
I could go on by I am shutting up now.
So please use the better term, Independent.