How many ranked games must I play to be ranked?

You will get placed into a League (Bronze, Silver or Gold, but not Platinum, Diamond, or Challenger) as soon as you complete all 10 of your "placement matches".

Near the end of the 10 matches, if you keep winning, the matches will include harder opponents -- presumably, to see if you are equal to them and should be placed into Gold. If you want to start higher, you can 'duo queue' with a friend with a high rank. [..assuming that you win :) ]

You need play at least 10 ranked games to be ranked. You will be unranked again when new season starts. Source Wiki.