The premise of Dogs in the Vineyard is that this is the Old West, more or less. You got trains, miners, Native Americans (though we call 'em Mountain Folk), and a bunch of small towns scattered across the land tryin' to get on. Most of these towns belong to the Faith. This ain't the case Back East, but out here, most of the towns are Faithful. Normally, a Faithful town goes along pretty well. There's enough rain, businesses go ok, nobody gets too sick. Sometimes stuff happens, but that's the way it goes. But, sometimes people fall off the right path. The first sin is always Pride -- somebody thinks they know better than they know, or they're better than they are, or a whole class of people is better than they are. Pride leads to Injustice. After Pride comes Sin. This is where things start getting bad, because when someone in town Sins, that opens the whole town to Demonic Attack -- crops start going bad, people get diseases, bad stuff starts happening. If nobody stops it, Sin leads to Corrupt Doctrine -- people start making up things about what the laws of the Faith are in an attempt to control the problem, and -- and that causes Corrupt Worship. When enough people adopt the Corrupt Doctrine, then you get a False Priesthood. That's getting scary, because at that point they're working with the demons, knowingly or not, and they got some powers -- we call that Sorcery. Finally, the False Priesthood leads to Hate and Murder, as sacrifice to the demons, to silence the opposition, or just out of revenge. At this point, the town's practically gone to Hell -- or at least it's practically under the controls of the powers of it.
So what's your job? Why, to fix all this, of course. See, every year, the King of Life calls some people to be Dogs. They go to the central city of the Faith, and they get trained up for a couple years -- they get taught to shoot and ride and read and write, if they don't know already; somebody makes them their special Dog's coat, and then they have their initiation. And then they're ready to ride. Dogs get sent in groups from town to town, doing marriages and baby-namings and settling disputes, carrying the news, and, most importantly, rooting out Sin. Dogs have the absolute and total right to dispense justice among the Faithful, both legally (at least by the Faithful's laws -- the country has some other laws, but whose do you think really apply in the towns of the Faithful) and morally, because the Dogs know the will of the King of Life. This is totally dictated by the player -- if the player says their character is acting in accordance with the King's will, they are, and if another player says theirs is, they are too, and if they're in conflict, well, they better work out what's up. Dogs can dispense justice anywhere from words of praise to shooting a guy in the street. Whatever's necessary to get rid of the Sin, because when Sin is gnawing at the roots of the town, eventually the whole thing is going to die.
The first page of the cheatsheet pdf is a pretty good rules summary, though it won't make a lot of sense on its own. Here's the basic deal beyond that:
At character creation, you assign some dice to your stats (four fixed values: Acuity, Heart, Body, and Will), to your traits (arbitrary) and to your relationships (with other people, NPC or PC, or to organizations, sins, places, or even demons), and to your equipment (in particular, you might want to assign to your gun, your Dog's coat, your bible, and your horse).
When a conflict comes up, we work out what the conflict is about-- "Does Brother Clarence get the mob to back down, or do they beat the hell out of him and the maybe-jinxed-the-crops-to-fail girl he's protecting?". Then we get to it.
The way the conflict gets resolved is, each person in the conflict takes turns being the leader. They raise -- put out two (already-rolled) dice and say what the total is and what they're doing. It has to be something their opponent can't ignore ("I have a 9 -- I go right up to the leader of the mob, and I say "I am the King of Life's Watchdog and this woman is in my care -- now go back home and take the rest of your boys with you!"). Then everyone else in the conflict has four choices.
They can concede -- the person who currently raised wins, and we skip to the bit where the conflict is resolved (every contest eventually ends with somebody conceding). They can see the blow with one die (that is showing at least a 9) and turn the blow. This means the raiser's action not only fails, but makes the situation worse for them ("I've got a 9 too. The leader of the mob takes a few steps back, then screws up his courage. "We ain't going nowhere, Dog," he says. "And I don't like the way you been ordering us around." The mob stirs angrily."). They can see the blow with two dice (that add up to at least 9) and block or dodge. This means they resist or avoid the raiser's action ("These two add up to 10. The leader leans in close enough that you can smell the bootleg whiskey on his breath. "Oh yeah?" he says. "You and what army?"). Or, the last choice, is they can see the blow with three or more dice (that add up to at least 9) and take the blow. This means the raiser's action succeeds, and the person who took the blow will be taking fallout later, so they should put the dice they used aside ("The leader winces at your words, and goes kinda pale as he tries to step back and blend back in with the crowd."). Note that if you can't, or won't, use enough dice to match the raiser's total, you have to concede. Assuming the raise was seen, then this 'round' is over, and the next person is now the leader. Eventually someone runs out of dice or decides it's not worth it to keep going, they concede, and the conflict is over.
When the conflict is over, the issue is decided. You can't start another conflict over it. You can, however, start another conflict over a related issue if you want ("Ok, you get the shit beaten out of you by the mob, and collapse unconscious for a bit. The last thing you hear before you black out is "Take the slut to the jail -- tomorrow morning we'll hang 'er." "Not on my watch -- I'm going to the jail and busting her out." "Ok, how do you want to do that?" "I'm going to scout out the territory, then sneak in past the guards." "Fine, let's roll it.").
None of this covers how you get dice, so here's the deal. When you raise or when you see, you can bring in your stats, your traits, your relationships, and your equipment. You bring in your stats by escalating -- going from one kind of conflict (talking, fighting, or shooting) to another. Depending on the conflict, you'll roll different traits. You don't have to escalate if your opponent does, if you want to talk down the guy with the gun or whatever. You bring in relationships by having conflicts that involve the thing you have a relationship to -- you can bring in your relationship with your brother if you're fighting your brother, or if you're fighting for your brother, or if you're fighting about your brother. You bring in traits and equipment by using them in your raise or see ("I pull out Ol' Betsy, my pearl-handled revolver, and let 'em see I mean business -- I'm rolling the dice for my revolver and for my I mean business trait"). You can even get dice for using items lying around, although generally they're not worth that much if they're not being used for their intended function and not listed on anyone's equipment sheet. You can only get dice for a particular attribute, trait, relationship, or piece of equipment once per conflict, but they all "reset" for the next conflict.
This also doesn't cover what the deal is with fallout. Basically, fallout depends on what the conflict type was (talking, shoving, punching, or shooting) and how many dice you used to take the blow. At the end of the conflict, you'll roll some fallout dice, and these tell you how much fallout you get. Low levels of fallout tend to mean you gain or modify your traits. Fallout is the only way to get new traits and relationships, or improve old ones. Higher levels of fallout may mean you take damage or even die. Remember that you can always concede a conflict instead of taking the blow if you're unhappy with how much fallout you'll have to take!