So the over-taxed, frazzled, and exhausted coalition said, "thank goodness," and let the Kurdish Peshmerga police its own territory. In the meantime, the US started taking shots at Sunni Arabs that were causing trouble (and some that weren't). Not only did this occupy many Sunni forces, but it chewed them up. This will become important.
The Kurds are a proud people that have faced a long history of oppression. After the First World War, they received vague promises from European powers that they would get a state. In the shuffle of great power politics, they managed to get themselves swept under the rug--Kurdistan never became a state. Instead, the region in which Kurds lived was broken up into parts of four countries: Syria, Turkey, Iran, and Iraq. Kurds found themselves ruled by Persians, Arabs, and Turks--none of which spoke their language and shared their culture. Attempts to gain full autonomy in one country met backlash from all four--Kurdish land is fertile and full of oil, and all four countries that contained Kurdistan knew that successful Kurdish independence from one state would lead to instability and increased nationalism at best, and momentum for a war of independence at worst. Hussein launched a war of genocidal terror against his Kurds after the Persian Gulf war, killing hundreds of thousands and prompting the US-led Operation Provide Comfort, which kept Iraqi troops from setting foot in Kurdsitan for years.
Provide Comfort was a defining moment in the history of Iraq's Kurds. Because Iraqi forces could not set foot in Kurdish areas, the Kurds had to look after themselves. Iraqi Kurdistan had actually gained official autonomy in 1970, but its parliament was under Hussein's control until the aftermath of Provide Comfort, when Iraqi forces left the area. Official autonomy had put most of the mechanisms in place necessary for real autonomy--without Iraqi troops (and with an economic blockade by Hussein), Iraqi Kurdistan had achieved de facto autonomy. The Peshmerga, a paramilitary force of the Kurdish region, policed the region. Two opposing factions ruled. It acted much like it's own country--similar to a Taiwan or South Ossetia situation. The Kurdish autonomous region is defined in figure 2 below; some Kurds lived outside the region, and a few Arabs lived within it.
The green areas represent Kurdish populations, but those blobs were certainly not static. In addition to genocide by murder, Hussein made attempts to consolidate Ba'ath control of Kirkuk and Mosul, the 5th and 2nd largest cities in Iraq (and quite wealthy) by a process of "Arabization:" moving Arabs into and around the cities, something reflecting "Hanization" by the Chinese in Tibet and Xinjiang. This process will also end up being quite important.
In the 12 years between Provide Comfort and the US-led invasion, the Kurdish population became quite used to its autonomy. The Peshmerga, a paramilitary force on the autonomous region's budget, kept the peace. The new generation of Kurds mostly doesn't speak Arabic, and largely does not have much interest in the Iraqi state as a whole--they see themselves as citizens of Kurdistan.
Enter again the Americans. During the Coalition's Invasion of Iraq, troops found themselves quickly overwhelmed with looting, murders, and then an insurgency. US troops, in charge of the northern zone of Iraq, asked the Kurds to keep the peace. And keep the peace they did--Kurdish troops marched south into areas they considered "Arabized" Kurdish territory, and a bit more. The US did show up in Kurdistan eventually, but not until the violence in Baghdad and Anbar started to fall. In the four years' meantime, the Kurds ruled. Their holdings have extended to those of the figure below.
Iraqi Arabs are not happy with this arrangement. Kurds have extended their control to northeastern Mosul and Kirkuk, Iraq's 2nd- and 5th-largest cities. Mosul is a wealthy trading city and Kirkuk a wealthy oil city--and both have lots of Arabs (interestingly, Mosul also has lots of Turkomen). While the Kurds managed to keep much of their territory relatively peaceful, their fighting with Arabs and Turkomen has turned Kirkuk and Mosul into 2 of Iraq's biggest violence hotspots. Mosul, in the confusion of the fray, has become Al Qaeda's last hideout in Iraq. It's a serious security problem.
Maliki sent the Iraqi Army to take back some of the Kurds' southeastern holdings that were pretty unquestionably Arab territory. There has been no civil war yet, but this is probably due only to the persistence of the US--the Kurds' best friends--in levying pressure to negotiate. But the Kurds are frustrated. Cities like Kirkuk were supposed to have referenda on whether they wanted to be a part of the Kurdish Autonomous zone or not--and the vote is now well over a year delayed, and shows no signs of happening. The US is trying to push for a negotiated settlement between the two sides. But Mosul and Kirkuk are likely to become embroiled in fighting no matter which way they go--even if their representatives can reach a deal, Arabs and Kurds in each city believe deep down that the city is theirs and that the other side has taken unjustly what they've taken. It's a mess.
The US doesn't have too long to stay in Iraq, and needs the negotiations to happen on a reasonable timescale--Maliki wants the same, because he knows that the US will back the Baghdad government if the Peshmerga picks a fight with them. The Peshmerga know it, too--but they also know that the US is willing to look out for their interests to some extent if they don't pick a fight with Baghdad--the US and the Kurds are still close friends with a relatively long history. But this explanation predicts that the Kurds would be happy to negotiate, and so far they haven't been thrilled. As the Iraqi Army grows, they will lose bargaining power--the only two rational approachs I can think of are A) negotiating now, while Kurd power is strongest with respect to the Iraqi Army and the Coalition, or B) wait until the Coalition leaves, at which point one would presumably have a boost in advantage. The latter is a dangerous game of brinksmanship that the Kurds must avoid. The former will probably not happen until an oil deal is passed in parliament--and that deal is getting close to 2 years delayed.
The resolution of this Arab-Kurd dispute when the US leaves will likely be the final step in securing Iraq's stability into the future. But failing to do it could undermine the progress that Coalition and Iraqi governments have made. Careful and persistent work awaits US diplomats in the next 2 years.