Mechanics. Math. To some people, these are the enemy. However, to a smart game designer, these are the tools and weapons with which she, he or they engages the game itself.
There are a lot of game designers, GMs, and players who view math and mechanical analysis with suspicion. "Metagaming," they mutter. "Roll-playing instead of roleplaying." What these snowflakes fail to realize is that the mechanics are how the players interact with the narrative. If you don't have your mechanics locked down, then your precious storytelling, roleplaying, and adventure is going to suffer for it.
An example: Let's say you're building a fantasy game where the players are supposed to be veteran soldiers and experts in their trades. For your resolution mechanic, you decide to go with 2d10, rolled twice and multiply the results. You figure that 170 is a pretty easy mark to hit. After all, it's out of 400.
Actually, 170 is about a 21% chance of succeeding on 2d10*2d10. The 50% mark is at around 112. So if you make your game with the assumption that 170 is a pretty good chance of success, your big damn heroes are going to fail most of the time, which means they aren't going to be very convincing as veterans who know what the fuck they're doing, now are they?
On the other hand, maybe you want a grim, gritty system where every fight has the potential for horrible death. But you fuck it up the other way, and now the PCs are nigh-on unkillable. The game does not support the story you're trying to tell.
There are tools you can use to fix this. One is playtesting, though god knows most game designers evidently have no time for that. Another is basic goddamned math. If you're just rolling d20+skill for soemthing, figure out what their skill should be, how hard it should be, and set the DC accordingly. If it's hard to figure out what their skill modifier should be, your skill system probably needs work.
Another tool you can use is Anydice (http://anydice.com/). That's how I found out 170 from 2d10*2d10 is a 21% chance (using output 2d10*2d10). You can get a dice pool (we'll say d10s, success on 7 or higher) with output [count {7..10} in 10d10]. For most dice systems, you're going to want to go with the At Least option selected, which gives you the odds of getting any particular number or higher. Play around with it.
Of course, the dice probability is just one piece of mechanics work. I mentioned earlier about skill systems not working right. This applies to basically any sort of modifiers you add to a skill, really.
If you don't keep a good handle on the modifiers a player can add to their skill, it makes it a lot harder to set reasonable target numbers. You want there to be enough swing to make sure that a character who specializes in something is better than one who doesn't, but you don't usually want to have the modifiers go so high they never fail (or rather, if they never fail at that task, it should be because that's the kind of character you want in your game). This can be accomplished by making sure you don't have too many stacking modifiers, and checking to make sure their bonuses don't go up too far as they increase in experience.
Sometimes, one mechanic might hamstring another. Sais are traditionally used for disarming attackers, and they're extremely good at this in real life. For this reason, they get a +2 to disarm in D&D 3.5. All well and good, except that sais are light weapons, and that gives a -4 to the roll. Even from the standpoint of realism it was a bad rule, because in real life, sais are a hell of a lot better than greatclubs at disarming people.
Check to see how different rules interact, and if something doesn't work the way it's supposed to, try to fix it. You won't catch every weird rules interaction, but if something is supposed to be good at, say, stabbing people, look at your goddamned stabbing rules and make sure it works the way you intend.
One last bit with skills. Skills, especially when they cost character creation resources (advances in Savage Worlds, skill points in Pathfinder/D&D) should all be things players are expected to do in a game, not just "well, this is a thing that people can do, I guess we should have a skill for it." Skills are one of the main ways players interact with the game world. Make your skill lists with an eye to what you expect players to actually do. If it's not something that the system actually wants you to do (for the most part, craft and profession fall into this category in Pathfinder), then don't have a skill for that, or else don't have it pull from the same pool of resources as the skills you actually expect them to use.
Because if a player sees craft (armor) in the book, and he thinks playing a blacksmith is going to be rad, he's going to be pissed if the game never lets him make a craft roll that actually matters.
These are just a few ways that your mechanics shape the game narrative. Every mechanic in the game is going to shape how the story goes in some form or another. It's going to affect the mood of the game, the agency the players have, and how much the game actually matches up to your expectations. Do the math. Think out the rules. And for the love of god, don't just assume shit's going to work. At least toss some dice around, if not have someone else playtest the fucking game.
