Open the Challenge page, or tap Single on the Play page. The Daily challenge has 5 steps and the Weekly challenge has 15.
Each step is a match against a computer opponent, and each attempt costs 1 ticket. Win the match to move on to the next step and collect its reward: coins, a ticket back and, on later steps, a chest. A loss or a draw keeps you on the same step.
The Daily challenge starts over every day at midnight UTC, and the Weekly challenge every Thursday at midnight UTC, whether you finished it or not.