This is different than rules-heavy rules-lite, but related. It's more like: how big is the library of useful-but-not-integral pieces with distinct mechanical meaning that are part of the game?
A really Skinny Game is rules-lite but, also, does not have a lot of moving parts. The classic example of a "moving part" here would be like a pre-written magic spell like Sleep. The Pool is a really Skinny Game. This superhero game is a Skinny Game. STACK is a Skinny Game.
A Fat Game would be like pretty much every edition of D&D. It has a library of spells that are all different, plus weapons, each with different damage (usually). The classes and races, each with mechanical differences, all fatten it up. Treasures and monsters fatten it up, but not quite as much because you don't have to use them and they aren't automatically part of the game. Plus every house rule and new bit on a blog.
Even though Call of Cthulhu is full of spells and monsters, I tend to think of it as relatively Skinny system since I never use any of the library stuff (except the guns) or maybe use one per game and they feel very much as if they're just there to make the book longer. Does Nyarlathotep need stats, really? Does explaining what "History" skill does genuinely count as content in Cthulhu?
Lamentations of the Flame Princess' lack of a monster list was much lamented, on account of--I think--less that people didn't want to make up their own or just use D&D ones but more that they expected any D&D derivative to have a certain amount of Fatness.
Rolemaster is Fat and crunchy--generally Fat Games are more crunchy than skinny ones but they can be relatively rules-lite like Marvel Super Heroes: it has loads or powers and skills in its library. If you are playing using the actual Marvel Universe as a setting, then the game is really Fat since the stats for all the heroes and villains are included in the game.
I tend to think of RIFTS as being so Fat it's almost Skinny--like, at a certain point you just go "Ok, you can do anything with this game, stop looking at the rules and just make shit up".
Indie games tend to be Skinny--inventing powers and abilities and details is usually part of the creative process that drives those games in play so there isn't a widget library to Fatten them.
EDIT--Important distinction: systems like Mutants and Masterminds and HERO where you build-your-own effects. I don't consider those effects as Fat as they'd be if they just gave you the thing fully formed. Not to say they are bad, simply that Fat is about straight-out-of-the-box library content, not tools to build content. Giving someone a novel is different than giving them the dictionary and going "You can build any sentence you want with this!"
A good Skinny game is elegant and makes you think about good game design. A good Fat Game is rich and inspiring.
A game like Vampire with tons of background that isn't tied mechanically to the gameplay could be thought of as either Skinny or Fat depending on whether you want to use that stuff, I guess.
Here's what I want somebody to write: A Fat fucking Game. A seriously cool new Fat Game full of big fat wonder. Fat as an obscenely baroque high art the way 1e Warhammer and Rolemaster did it. I want to see someone put in 99% perspiration.
Sleek little 3-paragraph flexible minigames--we have people on that. Every week there's a new one. Clever mechanics are getting produced and I trust they will continue to do so as the people who like to do that show no sign of getting bored with it and that important job is being done.
But right now what is going to get me excited is, I think, the Fatter the better. Getting pointed toward a genre and playstyle by a few good illustrations and an Appendix N is all fine and good but I can do that myself pretty much at will these days. I want to see something where nobody at the table knows what the fuck is going on and then you turn to page 546 and....whoa, I never knew that thing did that!
I don't need more crunch, but I want a million billion culled and polished ideas shoved into the fiber of the thing--when I invite the game over to play, I want it to be a multilayered and unpredictable player. I can and do invent stuff all the time, but I like a game that brings a lot of canned creativity to the table. It's fun to collaborate.
I'm supposed to be getting DCC in the mail soon--I hope it's Fat. I hope that the new D&D is Fat with stuff that counts as Fat because it wasn't anywhere in the earlier editions. But I hope no matter how Fat they are, some maniac is off somewhere at this very moment working on something even Fatter.