Quests are similar to the list of tasks you need to accomplish in The Sims Mobile. It is different from the Daily To Do List because Daily To Do List is only available until the end of the day. You can always continue your progress in the Quest as long as it is active, unlike with Daily To Do List that refresh the list of tasks every day. It's not necessary to complete the list because you can still do a lot of things in the game and you can always move forward.
Each Quest is normally divided into different stages. Each stage has a certain number of tasks and hints. By completing each stage of the quest, you will receive a number of Simoleons, XP or SimCash as a reward.
Where can I find it?
On the lower left side of the screen, tap the "Quests" icon.
Select "Quests" on top of the screen. You'll can see a number of quest you can complete anytime. You'll see its description and by tapping the encircled question mark, you can get hint on how to complete it. The reward for each quest is shown underneath the quest box. You can get Simoleons, Simcash and XPs.