Achievements are small, optional goals you can meet while playing the Altador Cup that can earn you extra Prize Points and exclusive prizes! They're small challenges that add to the difficulty of various games, but are not necessary to achieve in order to earn points for your team. They're just bonuses you can gain on top of the Prize Points and team points you are already earning.
Once you've completed an Achievement, you will be able to claim the reward for it on the main Altador Cup Tournament page. Achievement prizes can be claimed at any point during the tournament or after it, but must be claimed before the following year's Altador Cup, or they'll disappear along with the Prize Shop!