I am an indian with indian passport. My daughter is born in USA n has american passport. Do i need to apply for a visa for her if we are going to India? Also how it works if she is going to stay in India longer?

You are addressing this question in the wrong forum. This form is for Google Map users and we are not associated with the embassy. You would need to contact someone at the embassy

She must have a visa. If Tourist visa is available, it is valid for a year.

Yes

Get her Overseas Citizen of India card from embassy, you won't have any problem thereafter