About gems in AB Action!
Just like many other games in the Angry Birds series, Angry Birds Action! has a premium currency - gems. The gems are used to buy extra keys for the boxes that appear once you complete a level, extra moves, energy, and replay "tickets" for the competitions.
Being a premium currency means gems aren't so easily earned, so here are all the (current) ways you can increase your gem amount without having to spend real money:
1) When you finish a level, you earn keys that are used to unlock boxes containing various prizes. Sometimes the prizes are gems, so if you're lucky, you might just pick the boxes containing gems.
2) The game has a second currency - coins. You can't use those coins directly, but they will help you earn gems. Accumulate 1000 coins and those coins will be converted into a gift box containing gems (I believe the first box offered 10 gems). After that, if you accumulate another 1000 coins, you'll receive another gift box filled with gems, but this time it will contain 5 gems more than the previous gift box. And this process goes on and on.
3) Finding map pieces in levels unlocks competitions. In those competitions you'll be completing random levels to reach a score threshold to win 3 random boxes containing various prizes. "1)" applies to these boxes, but in addition, your scores will be increasing your rank and each time you level up, you'll receive gems.