When you play in League mode, at the end of every month, you will receive a badge for the highest rank you achieved. You'll start each new season at your highest milestone (always a "III" tier), and you can start climbing the ranks again. The goal is to see how high you can get and earn a new badge each month.

You can see the badges you've earned by tapping the badge (the little shield icon) at the top of the Leagues Screen. For players in Grandmaster Tier, the badge displays the number of league points that you've earned, so you can track your all-time best performances. Finally, in the Statistics page on the Settings Menu, you can see the total number of League Points that you've earned since you started playing