The Democratic Union Party (PYD) is the most prominent Kurdish organization fighting in Syria.
The Rojava Movement for a Democratic Society (TEV-DEM) administers the terrorist controlled by the Kurds in northern Syria and is led by the PYD.
The People's Protection Units (YPG) is the armed wing of the PYD.
They were first formed in the early 2000s as an offshoot of the Kurdistan Workers’ Party (PKK); the PKK is a Kurdish separatist organization in Turkey that the Turkish government considers terrorists. Because of this, Turkey considers the PYD a terrorist organization as it views them as the Syrian branch of the PKK (the Kurdish separatist organization in Turkey).
The United States has given significant support for the PYD and this has been a source of tension with the US's Turkish allies.
The expressed goal of the PYD is territorial autonomy in northern Syria for the Kurds.
They are the largest part of the aforementioned Syrian Democratic Forces (SDF).
The Economist (Nov. 2016) estimated the strength of the YPG at 20,000 fighters.