While Nintendo doesn't have a built in Achievement or Trophy system, that hasn't stopped some game developers of adding their own to each game. Pokemon X and Y are no different. There are a slew of Medals you can earn from doing just about anything in the game, but to start earning them you do have to set your game up properly first.
It all starts with Game Sync. In order to attain Medals, you'll need to go into your PSS menu and choose Game Sync, and then register your copy of the game to your profile on the Pokemon Global Link, using the 12 digit code at the top of the screen. Once done, you're ready to earn medals.
If you've already been playing the game, or beat it, don't fret, everything you've done up until now still counts.
Ace Pilot | Repeatedly use Fly to go places
Archeology Lover | Revive one Pokemon Fossil
Battle Learner | Fight 100 battles
Battle Maison 20-Win Streak (Double) | Win 20 consecutive Double Battles at the Battle Maison
Battle Maison 20-Win Streak (Multi) | Win 20 consecutive Multi Battles at the Battle Maison
Battle Maison 20-Win Streak (Rotation) | Win 20 consecutive Rotation Battles at the Battle Maison
Battle Maison 20-Win Streak (Single) | Win 20 consecutive Single Battles at the Battle Maison
Battle Maison 20-Win Streak (Triple) | Win 20 consecutive Triple Battles at the Battle Maison
Battle Teacher | Fight 200 battles
Battle Test Beginner | Get Beginner rank in the Battle Test
Battle Test Normal | Get Normal rank in the Battle Test
Battle Veteran | Fight 400 battles
Battle Virtuoso | Go through 2,000 battles
Beginning Trader | Link Trade one time
Berry Gardener | Pick 10 Berries from fields
Berry Picker Fan | Complete Berry Picker Unlimited
Berry Scientist | Successfully mutate a Berry
Broad Friendship | Register 30 friends
Bug-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
Busy Saver | Save 20 tmies
Capturing Spree | Catch 50 Pokemon in 24 hours
Central Kalos Pokedex Completionist | Complete the Central Kalos Pokedex
Coastal Kalos Pokedex Completionist | Complete the Coastal Kalos Pokedex
Countess | Get this rank in the Battle Chateau
Dark-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
Day-Care Faithful | Often leave Pokemon at the Daycare
Dragon-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
Duchess | Get this rank in the Battle Chateau
Duke | Earn the rank of Duke at the Battle Chateau
Earl | Earn the rank of Earl at the Battle Chateau
Easy Cycling | Ride a Bicycle 30 times
Egg Beginner | Hatch one Pokemon Egg
Egg Breeder | Hatch 10 Pokemon Eggs
Egg Elite | Hatch 50 Pokemon Eggs
Electric-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
Evolution Authority | Evolve 100 Pokemon, enough to astound Pokemon Breeders
Evolution Expert | Evolve 50 Pokemon and become recognized by breeders
Evolution Hopeful | Evolve one Pokemon
Evolution Tech | Evolve 10 Pokemon and become recognized by Breeders
Exp. Millionaire | Earn 1,000,000 in 24 hours
Experienced Saver | Save 50 times
Extensive Friendship | Register 10 friends
Fairy-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
Fashion Leader | Accumulate a lot of clothing
Fighting-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
Fire-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
First Encounter | Meet one other player in the game
First Friend | Register one friend
Fledgling Photographer | Send one photo to the Pokemon Global Link (PGL)
Flying-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
Frequent Trader Link | Trade 50 times
Friend Safari 10 | Get 10 Pokemon encounters in the Friend Safari
Ghost-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
Good Rod Fisherman | Battle 10 Pokemon while fishing
Grand Duchess | Get this rank in the Battle Chateau
Grand Duke | Get this rank in the Battle Chateau
Grass-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
Gratitude | Generously tip people
Great Customer | Spend $100,000 at PokeMarts
Great Trade-Up | Have 100 Link Trades
Ground-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
Happy Birthday | Get a happy birthday wish over the PSS
Hard Cycling | Ride a Bicycle 100 times
Hatching Aficionado | Hatch 100 Pokemon Eggs
Head It Fan | Complete Head It Unlimited
Heavy Walker | Take 20,000 steps
Honored Footprints | Take 100,000 steps
Ice-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
Indulgent Customer | Spend $1,000,000 in PokeMarts
Light Walker | Take 5,000 steps
Link Battle Amateur | Fight one Link Battle
Link Battle Pioneer | Have 10 Link Battles
Lumiose Explorer | Visit much of Lumiose City and become very stylish
Lumiose Star | Visit everywhere in Lumiose City and become very stylish
Lumiose Tourist | Visit a little of Lumiose City and become a little stylish
Marchioness | Get this rank in the Battle Chateau
Marquis | Earn the rank of Marquis at the Battle Chateau
Middle Walker | Take 10,000 steps
Mighty Fisher | Find 100 Pokemon while fishing
Moderate Customer | Spend $10,000 at PokeMarts
Mountain Kalos Pokedex Completionist | Complete the Mountain Kalos Pokedex
Naming Champ | Give nicknames to 10 Pokemon
Never Give Up | A Medal for those who don't know when to quit even when there's nothing they can do
Noneffective Artist | Mistakenly use a move that is not effective (i.e. Fighting on Ghost)
Normal-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
O-Power Lv. 1 | Use an O-Power one time
O-Power Lv. 2 | Use an O-Power 10 times
O-Power Lv. 3 | Use an O-Power 50 times
O-Power MAX | Use O-Powers 100 times
Occasional Trader | Link Trade 10 times
Old Rod Fisherman | Battle one Pokemon while fishing
Outgoing | Meet 10,000 other players in the game
Pedaling Legend | Ride a Bicycle 500 times
Poison-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
PokeMile Hoarder | Accumulate 10,000 PokeMiles
Pokemon Center Fan | Rest at a Pokemon Center many times
Pokemon-Amie Aficionado | Play in Pokemon-Amie 10 times
PokeRadar Operator | Get the data of 10 Pokemon using the Poke Radar
PR Video Debut | Show off your PR video once
Psychic-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
Regular Customer | Become a regular customer at a PokeMart
Rock-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
Smart Shopper | Buy 10+ Poke Balls to get a free Premier Ball
Sociable | Meet 100 other players in the game
Starter Cycling | Ride a Bicycle one time
Steel-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
Step-by-Step Saver | Save 10 times
Successor in the Making | Mega Evolve one Pokemon
Super Rod Fisherman | Find 50 Pokemon while fishing
Super-Training Instructor | Play in Super Training 10 times
Supereffective Savant | Repeatedly abuse type weaknesses (i.e. Fire on Grass)
Sweet Home | Return to your hometown
Television Kid | Repeatedly watch television
Tile Puzzle Fan | Complete Tile Puzzle Unlimited
Trash Master | Check trash cans repeatedly, despite them being empty
Trick Master | Use a variety of tricks on your Roller Skates
Viscount | Earn the rank of Viscount at the Battle Chateau
Viscountess | Get this rank in the Battle Chateau
Water-Type Catcher | Catch all Pokemon of this type in the Kalos Pokedex
Wonder Writer | Save 100 times