Al Jazira is a region largely between the Tigris and Euphrates rivers in the north of Iraq.
When travelling to Iraqi Kurdistan, avoid this province. This must be made extremely clear with taxi drivers (who will not be too keen on travelling to Al Jazira themselves).
Note that a HUGE error can be made if a foreigner accepts a ride from someone travelling to Kirkouk or Baghdad-- people from these regions are desensitized in some respects, and will drive around Al Jazira(through its suburbs) to shave off a half hour of driving time (by not driving around the Mosul province).
One can be aware if they are entering Al Jazira province by the flags at the checkpoints. Know the difference between Iraq and Iraqi Kurdistan flags-- it might save your life.