So, inevitably, people take apart games and the people who play them to classify them and what they like or (more commonly) dislike about them.
You get the whole GNS shebang which looks at people's 'creative agendas' and what they want out of the game, and stuff like that. Personally, I find that approach kind of lacklustre, as it doesn't tell us anything about the game's writing itself. Plus, the difference between simulating a genre's cliches and 'exploring the themes of a genre' is something I never really saw the distinction in.
Anyway, I tend to classify games into three broad types, based on the focus of their mechanics. Something about gamers, and particularly the ones that write their own games, seems to love dividing things into catagories, and I'm no exception there.
So, first up, we have crunchy games. These are the games where you have several big chunky books full of rules and interactions between rules. You often have a broad underlying mechanic, and then loads of modifications to it for different situations.
These games tend to put a lot of focus on character builds. What you can and can't do is strongly defined by the decisions you make in character creation. Often, a skilled optimiser can make a character that's much stronger than an unoptimised character. These sorts of games tend to involve a hefty character gen (maybe taking up a whole session or more) spent pouring through books comparing options, and then in play fairly strict adherence to the rules. In play, there tends to be a lot of focus on working the system in your favour, so that the raw mathematics of the situation rigorously defines what happens.
Often, these games are heavily combat focussed, but not always. Examples that I particularly like include the various Old World Of Darkness games. GURPS, 3rd and 4th edition D&D, shadowrun, the various 40k rpgs and loads more all come under this bracket.
Then, we have story games. These games put a mechanical focus on the actual plot, and typically give players tools to control who gets to control the narrative. Rules about who gets to narrate what, mechanics that control spotlight and tempo, collaborative storytelling and experimental games are all in this vague group. Games without GMs, or which give players a lot of tools to take over GMing fit in here nicely.
Usually, the mechanics are pretty abstract and universal. You tend to get the ability to come up with your own stats and powers, limited abilities to force the story in particular directions, and control over things that aren't your own PC. These games have much more of a tendency to think of themselves as Art, and to deal with genres that aren't variations on action, horror, fantasy and sci-fi.
Some of these games that I really like include Don't Rest Your Head, Monsterhearts and Lacuna. Other games you might have come across include Dungeon World (and all the other 'powered by the apocalypse' games), Fiasco, Dread, Fate, and way more indie darlings I've never played.
Lastly, there's rules-light games. These games tend to take the focus away from the rules themselves, treating them as an unfortunate inconvenience. Rules tend to not to be universal, and handle stuff on an ad-hoc case-by-case basis. Ideally, of course, these games don't want you thinking about the rules at all, and instead it's about doing smart things IC.
Character gen is usually quick, and often pretty random. Lethality might be quite high. Unlike story games, these games are pretty firm on the distinction between the GM and the Players: you say what you want to do, and the GM controls the results of your actions and literally everything else in the world. Unlike crunchy games, the game tends to rely on GM judgements more than rules - if the mechanics conflict with what the GM thinks 'ought' to happen, the GM takes control, not the rules.
Most OSR games fall in this category. Other games here that I particularly like include Paranoia and Trail of Cthulhu. To be honest, since this is my preferred style of play, I don't really know many games like this that I don't enjoy.
So there we go. Basically, it comes down to how you determine what happens next: strictly following the rules, using an abstract system to control narrative rights, or DM fiat.
Just my thoughts, and probably not massively useful.