A P.A. is the electronic system that's used to make announcements or play music in large spaces. Without a P.A. system for the talent show, the people in back won't be able to hear you play your flute.
The abbreviation P.A. is more commonly used than the full name, public address system. Once you start noticing, you'll realize there are P.A.s everywhere — in the supermarket (for tracking down lost kids and keys), in churches, at school, and in sports stadiums. Before the invention of the P.A., people had to amplify their voices using megaphones, hand-held cones that make peoples' voices project (slightly) farther.