The application for a PIO is done at the Indian Embassy, just as you would apply for a visa.
You will need proof of some sort to show that at least one of your parents or grandparents are or were Indian (I say "were", because it is alright if they have changed their country of citizenship). This can be in the form of Indian passports, Indian birth certificates, or school certificates. My father only had one school certificate (I forgot what level of education), and it worked just fine.
I'm not sure of all the legal and political advantages that come with being a PIO, aside from the advantages to students.
As a student studying in India, you need to register yourself at the city's Commissioner's Office, and get a Residential Permit (RP). The process of obtaining the RP takes loads of time and effort, since you will be having to travel back and forth between the city's head police station, your local police station, and your college for documents and signatures. If not done on time, you can be fined, jailed, or even sent back to your home country.
The RP must be repeated after expiration (I think it lasts one year? I can't remember).
With the PIO, it is not necessary to register for an RP, unless your stay is longer than 6 months. Usually, there will be a vacation within 6 months, so I have not yet had to register for an RP, since acquiring my PIO card. It works in place of a visa, and is valid for 15 years, whereas my student visa was only valid for 1 year.
There is the added advantage that holding a PIO card means I get asked less questions at the airport about who I am, what I am doing in India, etc.
On a side note, it has also comes in handy as a form of ID proof when opening accounts and whatnot. It counts as a second ID, when given along with your passport.
About the visas, I am not sure if this is exactly how it should be done, but I came here on a normal visitor's visa. After being accepted into the college, I converted my visa into a student visa, which is mandatory.
In order to convert, you must go to the Indian Embassy along with your passport and documents showing that you have been accepted into the college. This includes a Bonafide certificate, an eligibility certificate, and one or two more things that I cannot remember at the moment. If you live far away, I believe the embassy accepts mailed passports and documents. The embassy will simply cross off your visitor's visa, and put a student's visa on the next page. It costs some amount of money (I forget how much).
As far as I remember, it is easier to get these things done in your home country that in India. So you must come here, finish the admissions process, and either fill an RP or leave within the time specified by your specific visa, go back home, then either obtain a PIO card or have your visa changed to a student visa.
Hope this helps!