Most Christians in Iraq are Eastern Catholic. Iraqi jurisdictions include: Chaldean Catholic Church lead by Patriarch Emmanuel III Dely who resides in Baghdad; Syrian Catholic Church, which has an Archeparchy in Baghdad and another in Mossul; Armenian Catholic Church which has an Archeparchy in Baghdad.
Additionally, there are some 400,000 Christians in Iraq who belong to the Assyrian Church of the East lead by Patriarch-Catholicos Mar Dinkha IV.
More may be found at the Catholic Near East Welfare Association's website: http://www.cnewa.org/ecc-bodypg.aspx?eccpageID=1&IndexView=toc