Broadly, "Kurdistan" refers to the homeland of the Kurds, which straddles the borders of Iraq, Iran, Syria, and Turkey. The Kurds consider this to be their true nation. In practical terms, "Iraqi Kurdistan" is the Kurdish Autonomous Region in Iraq, with Erbil as its capital. It is not officially a country but is a sovereign state in all but name.
Up until 1991, the Kurds suffered terrible repression in Iraq under Saddam Hussein. All Iraqis suffered under Saddam's sadistic rule, but the Kurds got the worst of it, including the Anfal campaign and the chemical attack at Halabja.
In the wake of the Desert Storm war in 1991, there were uprisings throughout Iraq against Saddam's rule, by Iraqis who believed that the Americans would support them. They were tragically in error, and the uprisings were crushed by the government - except in the Kurdish north, where rebels managed to hold territory until the establishment of a UN-endorsed safe haven. Saddam's fall in the 2003 war allowed Iraqi Kurds, and the Kurdish Autonomous Region, to become full players in the region.
Iraqi Kurdistan is the only safe, stable area in Iraq, and one of the few such areas in the Middle East. There's never been a kidnapping of a Westerner here, and the region is untouched by the Sunni/Shi'a violence that has plagued the rest of Iraq. The terrorists of Daesh (ISIS) cannot reach here. The Kurds accept refugees (Kurdish and non-Kurdish, Muslim and non-Muslim) from the south, although this places a strain on the economy.
The Kurds do not consider themselves Iraqis - you won't see a single Iraqi flag here, unless it's officially mandated. Iraqi Kurdistan is a paradox: even though it's not recognized as a nation-state, it is more of a "real country" than Iraq itself. It isn't even a "state within a state" - it's a state within a non-state.