It's useful to write down certain choices when they seem to be mutually exclusive. This is an list of certain choices you can make, in particular choices that might be implicit.
What I mean by implicit is that when you have "an idea for a game" you might have already made these choices without realizing it.
They are all likely false dichotomies in some sense, but hopefully they are useful "design axes" to think about where your game sits.
Sorry no pictures today, I have bike riding to do!
On the one hand you have a game like Super Crate Box. Each of it's 3 levels does have a certain small narrative to it but that isn't why you are playing. On the other hand, a game like Dragon Warrior seems based around the process of exploring the world and a developing narrative and if anything the mechanics seem to be in place in order to structure or impede that exploration.
Mechanical games tend to operate on the principle of single screens, waves of enemies, or puzzles. I would categorize match-3s, sports games, tower defence, sim-types, world-building games or even RTS games as primarily mechanical. The actions you perform as a player are what is most interesting-- the focus is really on the player operation.
Narrative games tend to operate on the principle of immersion in a world, telling a story, or developing characters. Most RPGs are narrative games, as are dating-sim-style games, text adventures, action adventure games, and maybe even campaign-styled wargames (I haven't played any.)
One useful way to differentiate between them is to think: would multiplayer make sense here? If so, it's probably a mechanical game idea. Of course there are mechanical games where the answer is probably NO (The Incredible Machine) and narrative ones where it's YES (Secret of Mana) but there is something to this.
Another way to differentiate is to think in terms of what serves what: the backdrops in Super Crate Box clearly are there to make the game a bit more interesting, not really as your main reason for playing. Likewise, people HATE random battles in RPGs (which fortunately seem to be extinct now) but play them regardless. That's because the mechanics simply serve the narrative here, so something so badly broken isn't a game-killer.
Again: obviously there is a lot of crossover here, it's not a true dichotomy. The 3D and SNES Mario and Zelda games all do a pretty good job of marrying both elements, although Mario tends to be more mechanical and Zelda more narative. Likewise tactical RPGs, which have a strong mechanical element as well as a strong sense of world exploration or character development.
Does the player control only one character, or several (or many)?
There are lots of examples of single-character games, of any genre: virtually any FPS, zelda, mario, skyrim, pac-man, defender, and so on. Single-character games usually have the player "welded" to an avatar.
Multiple-character games are less common: tactical RPGs, other "classically styled" RPGs (such as most Dragon Quest after I, Final Fantasy, Ultima games.) RTS games would fall into the category, as would most tower defence games. If the player controls more than one character, they are less likely to be "welded" to any one.
It's possible to have a party-based game where the player is somewhat welded: Chrono Trigger works this way, by having a silent protagonist as well as a party.
I find this is an interesting choice for narrative-styled RPGs. Single-character games are, I feel, easier to design (or at least, easier to START designing) but a party-based game gives you more options for character and story development.
Does the world or game experience consist of a long string of places or events, or a gradually expanding domain?
This can be clearly seen if we compare for instance Final Fantasy 2, which is by-and-large an Odyssee, where the world is explored one location at a time, vs Zelda: LTTP, where the world unfolds in more of a outward-spiraling fashion as you gain abilities. All my game selections date me, but whatever! =)
The distinction here is sometimes made as "linear vs. non-linear" but I find this uncompelling. Many games people describe as non-linear are fairly linear (e.g., Super Metroid) so perhaps they are more reflecting on the way the world is built, vs. how many choices you have at any one time.
Consider: a match-3 game is highly nonlinear in it's core mechanic, since almost any move is usually legal, but is probably linear in other ways (level progression.) Unless it is a very special kind of match-3 it is likely Odysseal.
Of course, the Odyssee is the story of Odysseus who travels from place to place in a linear fashion. Each adventure, in turn, has little to do with the ones before or after. There is also a lot of goat-sacrifice, normally at the end of each chapter, which would be a good mechanic for leveling up in an RPG.
These are all very fundamental choices, and so by the time you've already conceived your game, you've likely already chosen. But it can be very interesting to go back and look at the choices you've made to understand why you may have (implicitly) made them.
Lately I've been doing just that, so hopefully in that sense this survey is useful to you, too.