Who are the Kurds:
The easiest way to answer this question is, of course, is to read the wikipedia article and I recommend that you do.
But, I'll also summarize and add my own info.
The Kurds are a people group living in modern Turkey, Syria, Iraq, Iran, Armenia and Azerbaijan. Due to various events in Kurdish history, there is also a sizable diaspora spread throughout Asia and Europe. There are also smaller communities in the US, Canada and Australia. If you read this blog regularly, you know there's a community in San Diego.
Their history is a complex mix of (mostly) Indo-European people groups moving into the area and adding their cultural, religious and language (and DNA, of course) to the mix. By 400 BC, the Kurds were known as Carduchi by the Greeks whose armies they attacked and they came under Roman rule in 66 BC.
In the seventh century AD, the Arabs showed up to conquer the Kurds. They, of course, were successful and most Kurds were converted from local religions to Islam.
The Kurds eventually became part of the Ottoman Empire and, after WWI, they found themselves inhabiting four new countries (and the Soviet Union) none of which had a Kurdish majority.
The Kurds have a distinct language from their neighbors, though years of minority status have influenced the language quite a bit. Arabic words can be found in the language of the Kurds in Iraq, Syria and even Turkish. Farsi words are common in Iranian Kurdish and Turkish has all but displaced Kurdish in Turkey.
In Iraqi Kurdistan there is an effort to rid the language of Arabic loan words and replace them with the Kurdish. The problem, in my experience, is that with some words, no one knows the Kurdish word. I remember my language helper calling his wife on multiple occasions and then listening as the struggled to come up with a non-Arabic word!
But, given the history of Arab oppression under Saddam Hussein, I can't judge their drive to purge Arabic from their language.
Where do they live:
Kurds live everywhere!
Just kidding, but, as I mentioned above, there is a large diaspora.
Kurds in Kurdistan are spread between Turkey, Syria, Iraq, Iran and Armenia.
In Iraq, there are three provinces - Dohuk, Erbil and Sulaymaniyah - which are recognized as Kurdistan. There is also a a majority Kurdish population in the oil-rich city of Kirkuk and the surrounding province. Article 140 of the Iraqi constitution promises a referendum on the inclusion of Kirkuk in the autonomous Kurdish region. This vote was scheduled for November 2007, but has yet to take place.
I have mentioned the mistreatment of Kurds in Iran, Turkey and Syria, so I won't go into it again now, but I will highlight a tidbit from the wikipedia piece on Article 140: Saudi Arabia reportedly offered the Iraqi Kurdish leaders $2 billion in exchange for delaying the process for ten years.
I mention that to highlight the fact that Kurdistan lays with hostile countries amid a hostile region. The Kurds have an oft-quoted proverb: "The Kurds have no friends but the mountains." In their history they have often been forced high into the mountains to escape persecution. Even today, it would seem that their neighbors work hard to destroy them.