Here is the full list of all 20 Duel of Summoners: The Mabinogi Trading Card Game achievements.
Challenge the Cat Merchant in a duel and win
Register a deck at the Draft Arena
Join 1 Internet Normal Battle
Clear Tutorial 1 and unlock Sarah
Start the Tutorial.
Make the first trade with the Cat Merchant.
Purchase booster for the first time at the Shop
Obtain 5 cards from a booster and obtain Sarah
Create a deck from 'My Card' and save
Claimed 'Starter Deck' from the Shop
Cleared up to Story Mode Episode 6: Doctor Blin Blin
Cleared up to Story Mode Episode 4: The Meeting
Cleared up to Story Mode Episode 3: But Elf
Player reached level 2
Started the game.
First time registering after starting the game
Sell 1 card to the Card Appraiser
Visit the Cat Shop and purchase 1 card
Clear all of Chapter 1 (10 Episodes) in Story Mode.