Duelling Club is a place where you can duel your fellow students and earn various prizes, including new powerful dueling spells. You can unlock dueling by completing Chapter 5 in Year 1, and can access the Duelling Room in the Dungeons location at Hogwarts (next to the Potions classroom).
To earn a prize in the Duelling Club, you'll need to win a certain number of duels. You can check how many wins you have and how many more wins you need to get to the next prize via the Quest Menu (scroll icon): choose the "Side Quests" tab and select the crossed-wands icon.