Ok so, I reference the narrative dice system a lot when I comment on various game related topics. Mostly because its what I've been running for the last couple of years, in the form of FFG's Star Wars RPG and the Genesys RPG. The system is a little confusing for newer players so I thought I would take a stab at explaining it.
At the core of the system is The Check. Anytime you want to do something where the outcome is uncertain, you make a check. To make a check you form a dice pool of both positive dice (generated by your characteristics and skill) and negative dice (generated by the difficulty and challenge of the task). The first is the Green Ability Die.
Ability Die
You get a number of Green ability dice equal to the higher number between the Characteristic and the Skill being used for the check. You have 6 Characteristics that make up the core of your character. These are Brawn, Agility, Intellect, Cunning, Willpower, and Presence. In Genesys characteristics typically have a value between 1 and 5. An average human in Genesys has a 2 in each characteristic. Skills represent focused training, and have a value between 0 and 5. It is possible to make a check with no ranks in a skill.
Proficiency Die
With your check you get a number of upgrades to ability dice equal to the lower number between the Characteristic and the Skill being used for the check. So if you have a 4 in a characteristic and 2 ranks in a skill, you upgrade 2 of the Green Ability Dice to Yellow Proficiency Dice. The same is true if you have 4 skill ranks and a 2 in the characteristic. These yellow dice have a higher chance of success and are the only die with the Triumph symbol. I'll explain the symbols in a moment.
Boost Die
Last for the positive dice is the Blue Boost Die. Boost dice represent small but meaningful positive effects to the check. Aiming, having an accurate weapon, getting assistance from another player, maybe an opening in the enemies defenses. Pretty much anytime you can convince the Game Master to give you a bonus to your check, you get a Boost Die. Now that you got your positive dice, lets get the negative dice pool going.
Difficulty Die
So checks will have a difficulty ranging from Simple (No Purple Difficulty Dice) to Formidable (5 Purple Difficulty Dice). Outside of combat, the difficulty is determined by the Game Master with most checks are between 2 (Average) and 3 (Hard) Purple Difficulty Dice. Sometimes you will be competing against another character, which can be both PC and NPCs. The difficulty in this case is determined by the opposed character's positive dice pool with Purples instead of Greens and Reds instead of Yellows.
Challenge Die
The Red Challenge die represents when something is not just difficult, it is challenging. Similar to the Yellow Proficiency it is an upgraded difficulty die making failure more likely. The Red Challenge Die also has it own special symbol, the Despair. Most of the time these only show up when the baddie is more powerful or a story point is spent by the Game Master to upgrade the check.
Setback Die
Lastly is the Black Setback die. These are used for various negative environmental effects, like darkness, the defensive qualities of armor, disorientation, or fear. Basically these are the little things that aren't going your way for the check. So now you rolled a fistful of dice, what does all this mean. Lets take a look at an example pool.
Example Dice Pool
This pool has 2 Proficiency Dice, 2 Ability Dice, 2 Boost Dice, 1 Challenge Die, 2 Difficulty Dice, and 1 Setback Die. So the first thing we count up are Successes and Failures. Successes look like the eight pointed stars and failures are the Xs. There are a total of 6 successes and 3 failures, as the Triumph (the Success with a a circle around it) and the Despair (the failure with a square around it) each provide a success and failure respectively. For every failure we cancel a success, leaving us with 3 Successes total. As long as you have 1 uncanceled success, the check succeeds!
Second step is counting Advantages and Threats. Advantages look like arrowheads and Threats look like a circle with three lines. So there are 5 Advantages and 2 Threats, so also cancel each other leaving 3 uncanceled Advantages. Finally you count Triumphs and Despairs. Unlike the other symbols, they do not cancel each other. The final result is 3 Successes, 3 Advantages, 1 Triumph, and 1 Despair.
And now you know how to use the Narrative Dice system. What I like about the system, is with the Advantage and Threats, is it not just pass/fail. You can pass a check, but have bad stuff happen with Threats or you can fail a check but use Advantages to help the situation. The dice system allows for a lot more nuance and you can use the dice to explain why things happen. Like if attack in darkness and the setback dice show failures, you missed because of darkness.
So that's it, does my explanation make sense? Let me know in the comments if you have any questions. Until next time happy gaming!