This has no simple answer, because ethnogenesis is a highly complex matter. It is generally accepted that in Middle Persian, "Kurd" meant something like "tent-dwelling seminomad" (for example, in the Karnamag-i Ardashir, Ardashir is raised by "Kurdish shepherds"), in particular nomads who spoke Iranian languages. In the days of old, the area of Kurdistan and its vicinity would have been populated by pastoral peoples speaking a variety of Northwestern Iranian languages existing on a continuum (that is, two dialects close to each other might be nearly indistinguishable, but the further away you go, the more different they get). The region saw an influx of Turkmen in the Middle Ages, leading to the emergence of Azeris on one hand as Turkish displaced local dialects, and Kurds as those who retained Northwest Iranian dialects.

Historically, "Kurds" would absolutely not have identified as a single people, but instead according to tribe, clan, kinship, allegiance, and lifestyle. Nor can their ancestors be pinned down to any one group (e.g. the Medes), nor can it even be said that their ancestors were entirely Iranian, since many other peoples (e.g., Armenians, Hurrians) lived in the area at various times, and would inevitably have intermingled with speakers of Iranian languages, as happens when populations reside near each other. As with countless other ethnic groups (the classic example is that French only became the majority language in France in the mid-19th century; before that, people spoke regional languages very different from the court tongue), Kurds truly emerge as a group as the result of modern-era nationalism.

I don't know of any one great reference work on this complex and highly charged subject, so my best recommendation is to peruse the various articles on Encyclopaedia Iranica for further reading on the subject.