There are many words associated with the place Philippines:
You have - Philippines, Filipino, Filipina, Pilipino ...
What is with all the spellings!
So here is how it is:
Philippines was earlier called : Las Islas Filipinas thanks to this dude whose name in Spanish was Felipe II de España. After a while the Americans landed and figured - "hey! the country's name is spelt wrong .. in our .. umm .. mother tongue(?), its spelt "Philip" .. So then the country's name became Philippines.
But by then, the people born in the Island of Finipinas were already called Filipinos. This worked well for the race which could make the 'F' sound. But the locals couldn't really pronounce F.. The max that came out of their mouths was a P. (I'm sure you've been asked if you would like some Prench Pries to go...? or Puck you...?). So that explains Pilipinos.
Now they really needed a language that all of Philippines could call 'National'. So then came the language Filipino which was basically Tagalog + (more stuff from the world). The letter F is a part of the language Filipino but not a part of Tagalog (Even if no one bothers to pronounce it). So that explains Filipino.
Women didn't feel sexy enough when they were referred to as Filipino. So they figured that they were to be given a separate oomph factor. And that explains Filipina.
So in summary we have:
Philippines - the country (pronounced with a F sound if you can .. else P sound)
Filipino - a male from Philippines (pronounced with a P by the locals and F by whoever can make F sounds)
Filipina - a female from Philippines (pronounced with a P by the locals and the world has no idea about this word)
Filipino - the national language of Philippines
Pilipino - is that mispronounced word which everyone understands.
- humble me