Kurdistan is a region named after the Kurdish people in the Middle East, divided between Turkey, Syria, Iraq, and Iran. The Yazidis are an offshoot of the Kurds, speaking Kurdish but still practising a version of the pre-Islamic Persian religion.
Kurdish is an Indo-European language closely related to Persian. It has three major dialects: Kurmanji or Northern Kurdish, spoken mostly in Turkey, Syria and Armenia, Sorani or Central Kurdish, spoken in most of Iraqi Kurdistan and parts of Iranian Kurdistan, and Pehlewani or Southern Kurdish spoken mostly in Iran. These dialects are not immediately mutually intelligible with each other. The script of choice for written Kurdish also differs according to the country — the Roman alphabet is in use in Turkey, Syria and Armenia while the Kurds of Iraq and Iran use the Arabic script.
Zaza or Zazaki is a related language spoken in parts of Eastern Anatolia in Turkey. While from a linguistic standpoint, Kurdish and Zaza are distinct languages, many native Zaza speakers identify with the Kurds.
Most of the locals are bilingual in the national language of the country they live in: Turkish, Arabic, or Persian. The Kurds of Armenia are usually trilingual in Kurdish, Armenian and Russian. There are also minorities with different native languages in the region, such as the Syriac.
Much of the region is prone to ethnic strife, political conflict or is a war zone: do your research well before attempting any visit.