Iraq has a serious problem with the Kurds. OK, that's an oxymoron, as Iraq has had problems with its Kurds since Iraq was cobbled together from parts of the defunct Turkish empire in the 1920s. One of the three provinces, the northern, largely Kurdish one, was originally part of the Turkish homeland. Iraq got it because there was oil there, and Britain did not want the new country of Turkey to have oil, just in case the Turks decided to try and put their empire back together. The other two parts of Iraq were former Arab provinces (Baghdad and Basra). The problem is that the Kurds are not Arabs and want to rule themselves. They have never been able to achieve that goal. The Kurds are particularly hostile to the idea of being ruled by Arabs, who they despise (the feeling is largely mutual, as the Kurds are Indo-European, not Semitic like the Arabs).
The Kurds are only 20 percent of the population, and until the 1990s, were kept down by Arab soldiers, who were more numerous and better armed than the Kurdish clan militias. But after 1991, when U.S. and British air power kept Saddam's troops out of the Kurdish areas, the Kurds built an army. U.S. Army Special Forces were sent in to train the recruits, and the Kurds learned well. Now there are 200,000 Kurdish troops, trained and organized as combat units. There are over 100,000 additional armed men available as a sort of reserve. American trainers noted that they got much better results with Kurds, than with Arabs.
This makes the Iraqi government nervous, because the largely Arab Iraqi Army only has 250,000 troops. There are another 250,000 paramilitary troops and police, but they are no good trying to pry the Kurds out of their mountains. Iraqi Arabs have seen the Kurdish troops in action, as the Kurds have sent contingents down to help fight terrorists. The Arabs couldn't help but notice that the Kurds had absorbed their American training well. The Kurdish troops were more difficult to bribe or intimidate, and these Kurdish fighters didn't like Arabs. All this has led the Iraqi Arabs to conclude that their best strategy is to try and negotiate a deal with their Kurds. But the Kurds want control of all, or some, of the northern oil fields. Then there is the issue of stolen property, which happened in the 1990s, as Saddam drove more Kurds out of their homes, businesses and farms (and let the survivors flee north), and replaced them with Arabs from the south.
But at the moment, the Kurds are willing to go along with the fiction of a united Iraq. Two divisions of Kurdish troops are serving under the command of the Iraqi Army, to help suppress Islamic terrorism in the north, especially around Mosul. Another 30,000 Kurdish police men, mostly in all Kurd units, are also helping with security in the south. These 60,000 Kurds could leave Iraqi government service quickly, instantly becoming a big problem, not part of a solution.
Meanwhile, the Iraqi Army has come a long way since 2003, when the old, Sunni Arab dominated force was disbanded, and a new one, loyal to a democratic government, and led by newly recruited and trained officers, was built from scratch. Because the Sunni Arabs loyal to Saddam (and Sunni Arab rule) fought a four year terror campaign, while the army was forming, the best troops were concentrated in special "intervention" units. This resulted in an army organization consisting of one "Intervention Corps" and three other corps of lesser quality. Total manpower is about 250,000 troops. Most divisions have four brigades, and a total strength of about 12,000 troops.
The 1st Intervention Corps consists of the two motorized, one infantry and one armored division. One of the motorized corps is the also known as the Reaction Force Division. This is considered the most effective division in the army, and one to be used for the most difficult situations. Think of this corps as the new "Republican Guard." The other three corps are named after the part of the country they are based in.
The Northern Corps has two motorized divisions and an infantry division. Two divisions of Kurdish troops are now under army control as "mountain divisions".
The Central Corps is the area around Baghdad, and the thinly populated Anbar Province to the west. This corps as the 1st and 2nd Presidential Brigades (for guarding senior officials). The Baghdad (56th) Brigade (for assisting with security in the city), two motorized divisions, one infantry division and one commando division.
The Southern Corps has three motorized divisions, with an infantry division being formed.
There are other security forces, mainly four divisions of Federal Police, the Counter-Terror Command (with seven commando battalions and support troops), the Border Police and half dozen battalions worth of "Emergency Police" (SWAT, riot control) distributed around the country. These other security forces are nearly as large as the army, but are not as heavily armed, or trained for heavy combat.
The navy is currently, basically a coast guard. The air force consists of about a hundred transports, helicopters and recon aircraft. The remaining U.S. troops are mainly deployed in bases around Baghdad, and northern cities.
Even the army does not have a lot of heavy weapons. There are only about 300 old T-72 tanks, and lots of newer armored hummers. Thousands of new armored vehicles are on order. Compared to Saddam's force, the NCOs and officers (mostly from the Shia Arab majority), have less time in uniform, but are better trained. The Iraqis have learned a lot about fighting from their American mentors. For a long time, Iraqis were considered the most ineffective military forces in the region. Not so much, not anymore.