Task vs. Conflict resolution is one of my favourite subjects, because it's straightforward, easy to explain, and immediately useful to design and play. This came up in a post on the Barf Forth Apocalyptica forums, and I thought it would be worth sharing more broadly, if only to provide me something to link people to, rather than swamping threads all over the internet.
Here's a task: "Swing my sword at a guy"
Here's a conflict: "Kill a dude"
Here's a task: "Drive my car real fast through the city"
Here's a conflict: "Get to the hospital in time to save my buddy"
Here's a task: "Climb in through a window"
Here's a conflict: "Get into the tower without being seen"
Tasks are what your character is doing. Conflicts are about what your character wants. It doesn't matter what scale your conflicts and tasks are at. Tasks are what the characters are doing - what's happening in the fiction. Conflicts are about the outcome - what do we care about?
All instances of play have task resolution. A character does a thing, and there's a way of finding out whether they do it well or badly. That system could be explicit in the text, or it could be implied (usually they're explicit though). The system could use some kind of randomiser or other resolution "Roll a d20, add your skill, and try to beat the number the GM says" "See if your Skill score is high enough for the task you're doing, and if it is, you do it well" or it might not: "Say how well your character does a thing. That's how well they do it".
I say "all instances of play" because although the text of the game's rules might not say anything about it, as soon as you have fictional characters in your game performing any kind of action, you have a system for deciding how well they do them.
All instances of play have conflict resolution. A character wants a thing, something is in the way, and there's a way of finding out if they get what they want or not. That system could be explicit in the text, or it could be implied (in traditional games, they're often implied or absent). The system could use some kind of randomiser or other resolution "roll a d10, add your strength. Subtract that number from their hit points. If they've got none left, they're dead" "See if your Will is higher than theirs. If it is, they tell you what you want to know" or it might not: "See how well you did the task, and then the GM will tell you whether that got you what you wanted or not"
That last example is the most common kind of conflict resolution in traditional games. Call of Cthulhu, AD&D (outside of combat), World of Darkness (mostly), all of those pretty much exclusively have systems where the dice tell you how well the character does a thing, and then it's entirely up to the GM whether that gets them what they want or not. People call those "task resolution" games, but in play they have conflict resolution too, it's just handled almost entirely by the GM.
A common thing GMs in those games do is to make conflict resolution depend on task resolution. They'll say "If you make your driving roll, you'll get to the hospital on time" or "If you make your climbing roll, you'll get into the tower without anyone seeing". That's not written down anywhere in the rules text, it's just a thing a lot of people have learnt to do.
Another thing a lot of GMs of these games do is make their decision dependant on their sense of drama and timing "Ok, you've succeeded on your roll, so you're driving fast through the city, and you're nearly at the hospital. But your buddy in the seat next to you, he's burst his stitches and he's bleeding out. You're gonna need to put pressure on the wound and drive at the same time!"
One more thing some GMs do is make their decision based on the needs of the story they have planned in their heads, or that they're imagining the players want. "You're driving fast through the city, you don't hit anything, and you pull up outside the hospital. You run around to the other door to drag your buddy out, but by the time you get there it's too late. He whispers one last thing: "Project... ...Manticore"
So that's one way of doing conflict resolution. Now lets look at an explicit (as in, contained in the rules text) conflict resolution system. D&D3E.
I know. You're all like "But!" and I'm like "Deal with it, haters." D&D3E has an unambiguous conflict resolution system. It's just really difficult to use, time consuming and fiddly, and frequently ignored in play. Except for combat. Combat is the most straightforward conflict resolution system in the game. Let's take a look:
You want to kill a thing, so you take out your sword and hit the thing with it. That triggers the combat conflict resolution system. There's some fussing with initiative and such, but the core of the system determines two things: How well you hit the thing with your sword, and whether or not that kills it. That's task and conflict resolution right there:
"Roll a d20, add your strength and attack bonus, plus any other modifieres, and try to get over the target's AC" This is task resolution. Do you hit it with your sword?
"If you hit, roll a d10, add your strength, and subtract that number from the target's hit points. If they've got zero or fewer left, they're dead." This is conflict resolution. Have you succeeded in making them dead?
All right? Now lets look at non-combat conflict resolution. Your dude wants to climb up a tower into the window, and he doesn't want anyone to see. The DM has prepped this area, so we know where the guards are and how high the window is off the ground, and all that. Lets assume right now that the guards are all elewhere, and all we need to worry about is getting in that window. The system says (and I'm paraphrasing right now because I don't have the books with me) "Roll your climb skill against a difficulty set by the DM. The difficulty is 15 for masonry walls, +5 if it's slimy or wet (which it isn't, according to the GM's prep). If you succeed, you move your speed score in feet up the wall. If you fail by less than five, you move half your speed. If you fail by five or more, you fall."
We roll the dice, you get a 20. So your dude moves 30 feet (your speed) up the wall. That's task resolution.
Is that high enough to reach the window? We look to the DM's prep. If so, you're in, if not, you have to roll again. That's conflict resolution.
In practice, most groups don't rigourously apply the conflict resolution rules in D&D3E, and end up leaving conflict resolution entirely up to the DM, who uses one of the process I described above. D&D3E conflict resolution requires immense dedication to following a lot of fiddly rules, as well as detailed and comprehensive prep from the DM. It's a huge pain in the ass, and hence, in my opinion, not a good set of rules.
Apocalypse World has an explicit conflict resolution system because if you follow the procedure set out in the books, conflicts between fictional characters will be resolved, without the decision being left up to the MC or to any player. Sometimes the MC will have to make a judgement about the fiction of play (will they suck it up, or give in?), and sometimes the player will have to make a decision (inflict terrible harm, or take little harm?), but those aren't usually in and of themselves deciding the outcome of the conflict. Sometimes conflicts won't be resolved by a single move. Sometimes there'll be a whole snopwball of moves before a conflict is resolved. We don't always know whether a given move is going to resolve the conflict right now or not. Sometimes it will be left unresolved for a while, but the immediate situation will have changed. This is what makes Apocalypse World different from explicit stakes-setting games like Dogs in the Vineyard or PTA, and more similar to other conflict resolution games like Sorcerer or D&D3E.