It was founded, initially, with funding from congress, as an independent, non-profit broadcaster. Currently, NPR receives very little to no funding from the Government. Something like 2% of NPR's funding comes from state, local, and national government. Though individual stations may receive radically varying amounts of government funding.
It is Public in the sense that it serves the public interest and is funded by listeners.
"According to the 2009 financial statement, about 50% of NPR revenues come from the fees it charges member stations for programming and distribution charges."[0]
NPR is not state media. That would be Voice of America, which is not allowed to broadcast in the US.
There is a good article here on its status:
http://www.npr.org/yourturn/ombudsman/2001/011228.html
"Legally speaking, NPR is a private, not-for-profit corporation chartered by the District of Columbia and qualified by the Internal Revenue Service as a "501(c)(3)" organization exempt from taxation. "
*edit - now seeing all the answers below