Milestones constitute a mission-esque reward-based system by which players complete certain objectives in exchange for rewards (think Gems, Ingots, Soul Shards, etc.). There are 2 main sections of milestone: Daily and Milestones. Within the Milestones section there are 4 subsections: Story, Event, Training and Other. All milestones and their rewards may be accessed from the game's entry screen, with the milestones button placed in the top-left section of the screen.
It is, of course, highly recommended to reach as many lucrative milestones as possible - especially those which award Gems, as they are among the main sources of said Gems. The rewards they provide are not to be overlooked and can easily aid you in reaching higher levels of gameplay.
Note: There is a limit of 20 milestones displayed within a category that have not yet been completed. If you have trouble finding a milestone, it may be because you have too many. You can still complete the milestone and then you will be able to see it. You can look at the event pages on this wiki to find a list of milestones. In addition, some milestones only display when you have completed the previous one for the same quest.
Every day there are a series of daily milestone quests available- this is where the bulk of your daily AP will come from as well as other rewards. See Daily Milestone Quests for a list of all the currently available dailies.
There are milestone rewards available for completing each story quest for the first time. These rewards can provide a significant source of gems and player experience as well as other rewards. See Story Milestones for a list of all currently available story milestone rewards.
For most events there are a set of milestone rewards available specific to the event. These are usually only available while the event is ongoing. If an event is repeated later you will not be able to get credit for any event milestones you already completed on the rerun, but your progress on the milestones will be remembered for the repeated event.
These milestones are typically rewards for achieving a certain player level (one is available for every 5 player levels starting at level 15) or leveling up a unit, a unit's job or evolve it a certain number of times. You can view a list of Player Level Milestones.
Sometimes daily milestones for collaborations or special event milestones may appear in here. Previously there were Monday missions that appeared here every Monday, but they are no longer available.