The Battle of the Philippine Sea is mission 8 in the American Campaign. You personally need to shoot down 20 planes with aircraft or AA guns. Kills that occur while you are looking at the Tactical Map do not count for this achievement. There are a lot of enemy planes in this mission and if you are playing on Veteran there is not much chance of winning without earning this achievement. Start with a CAP of Hellcats and kill the incoming Kates and Zeros at the beginning of the mission, and then flying boats will attack from the north. Between those two groups you should be very close to 20. There is also an enemy CAP over each carrier and your fleet will be periodically attacked by land based bombers from the east.