Medals are earned for completing certain challenges. They give boosts such as increase defence, attack, luck and several others. You can only wear one at a time. Usually, harder challenges yield better medals.
Fullscreen capture 2072012 12012 p.m..bmp

Medal Page

The Medals

Medals Required Challenges Boosts
Bandaid Badge Splat! +6 Defence
Medal Of Pain Medal Of Pain Too Close For Comfort,Oops Wrong Button, Damager, Meat +15% Attack
Overkill Trophy Overkill Trophy Instakill!, Overkill!, Annihilation!

+10% Attack

+10% Luck

Terraformer Certificate I Hate Boxes, Boxageddon, Scorched Earth +15% Bigger Explosions
Underdog Badge Not so Pretty Now, Scavenger, Down To The Wire +20% Coins
Flame Badge It Burns! +6 Attack
Eagle Eye Badge Careful Calculations, Eagle Eye, Aiming Is For Suckers, Artillery Expert. + 25% Knockback Resistance
Marine Certificate Master Arsonist, Out Of The Frying Pan..., Glub Glub, Let Me Help You +15% Jump Power
Trap Master Trophy Beep Beep Boom!, Take That!, Hidden Dangers, Close Call +8% Speed
Creativity Medal March of the Boxes, Box Angry!, To Me My Minions, Terrible Boat +50% Damage to Objects
Sharpshooter Trophy Nailed It!, Sniper, Bullseye, Two Birds With One Stone -20% Shot Spread
Trophy of Wealth Hooray Money!, Filthy Rich, Ridiculous Lucre +10% Coins & Xp
Trophy Of The Veteran Down In The Trenches, Front Lines, Veteran, Lean Mean Fighting Penguin

+10% Attack

+10% Defence

Ribbon Of Expertise Learning The Ropes, Hunter, Growing Up, Standing With The +20% more Xp
Mark Of The Assassin Assassin, Ravager, Eviscerator

+50% Attack

+50% Accuracy

Insanity Medal Variety Is The Spice Of Death, Armageddon, Penguin-Fu, Overeater

+75 Attack

-25 Defence

Threads Of Fate Medal Twins!, Master Warrior, Nick Of Time, Wardrobe Malfunction + 100 Luck