It's a kind of Hindustani classical song called a "khayal," sung, I gather, in Raag Jogiya. At the end of it she says, famously, "My name is Gauhar Jan!"
Who was Gauhar Jaan? Her background, from what I've been able to find on the internet, seems remarkable:
Gauhar Jaan was born as Angelina Yeoward in 1873 in Patna, to William Robert Yeoward, an Armenian Jew working as an engineer in dry ice factory at Azamgarh, near Banaras, who married a Jewish Armenian lady, Allen Victoria Hemming around 1870. Victoria was born and brought up in India, and trained in music and dance.
Within a few years in 1879, the marriage ended, causing hardships to both mother and daughter, who later migrated to Banaras in 1881, with a Muslim nobleman, 'Khursheed', who appreciated Victoria's music more than her husband.
Later, Victoria, converted to Islam and changed Angelina's name to 'Gauhar Jaan' and hers to 'Malka Jaan'. (link)
Through her mother, who depended on the patronage of wealthy Muslim noblemen (I'm presuming she may have been a Tawaif), Gauhar Jaan got training from the best classical music masters in Calcutta at the time. By 1896, she was a star performer in Calcutta, which is how she was able to charge Rs. 3000 in 1902 to have her voice on the first audio recording of an Indian song ever made. Later, Gauhar Jaan became a star all over India. She performed in Madras in 1910, and even performed for King George V when he visited India. She died of natural causes as the palace musician of the Maharajah of Mysore in 1930. (There is a fuller bio of Gauhar Jaan here, at the Tribune. Also, see this profile of Gauhar Jaan.)
Another song Gauhar Jaan was famous for was "Ras ke bhare tore nain," which I think many readers will find familiar for reasons that will become apparent below.
Here is a somewhat more recent version of "Ras ke bhare tore nain," sung by Hira Devi Mishra (from the 1982 film "Gaman"):
I'm finding the Hindi (Braj Basha?) a little hard to follow, so if anyone wants to help with translation, it would be appreciated. Here is the Midival Punditz' "Fabric," a drum n bass remix used by Mira Nair in Monsoon Wedding:
The neighborhood where she films those crazy wires is in Old Delhi -- the area around Jama Masjid. Nair also did her first, student film in that neighborhood (the film was her thesis at Harvard; it was a short, eighteen-minute documentary called "Jama Masjid Street Journal").