Events are weekly side games that are played separately from the main game for a limited time. Players must first be at Rank 8 or higher in order to participate in events.
There are currently 12 events in total: Agency Anniversary, A Very Merry Celebration, Awesome Architecture, Aztec AdVenture, Compelling Careers, Egypt Extravaganza, Impressive Infrastructure, Incredible Innovation, Japan In Jeopardy, Musical Mischief, Mythical Monster Hunt, and Scandinavia Scramble, with more on the way sooner or later.
One important part of doing well in the events is to be active. It's important to start the events as soon as possible after they begin and play often enough to claim mission rewards, invest resources & people, upgrade heroes and claim free capsules. If population missions take longer than production missions, it's best to stay on the event to claim Time Pods for more people. If production missions take longer, it's best to be off the event until the missions are done (provided that the bottom generator in the relevant age(s) are population limited) because offline production is calculated differently than when online, and in a way that gives you more resources. Some events have the same type(s) of missions the most difficult the most often. In those events, prioritize the heroes that help with those missions. Specifically, most normal and long events have the second Age generally having the most difficult production missions, the exceptions being those which do not have x * 3^lvl (or x * 3^lvl + lvl^2) common bonuses.
There is a Mission Tracker owned by Engima that you can use to plan ahead regarding when to upgrade heroes (for spend EXP or level up cards missions) and when to open free capsules (if a collect cards mission is coming soon, save the free capsules for the mission unless they will be maxed for too long by doing so). You can also use its calculator function to determine how long other types of missions will take, which is very useful if you want to take advantage of the offline production algorithm and complete missions as quickly as possible. Since capsule rewards increase with each rank, you may want to save free capsules and big mission capsules for the next rank if doing so is feasible.
There are 3 types of events: short, normal, and long.
Short events include Innovation, Infrastructure, Architecture, and Careers. These events are easy and can be consistently completed with little to no spending. In Innovation and Careers, it is important to be wise with your EXP spending, especially if you spend 50 gems or less on EXP, since capsule EXP is low and there are many EXP-related missions. These events last for 52 hours or 76 hours and have 15 or 20 ranks.
Normal events include Japan, Mexico, Scandinavia, Music, and Egypt. 2 of these events (Egypt and Japan) need a wildly varying amount of gem spending to complete, depending on RNG. Music cannot be completed without lots of warps and/or bundles. The best you can reasonably do is one mission away from finishing, and even that's not very feasible. Completing any of the 3 missions at that point to finish the event takes more than 24 hours. Mexico cannot be completed without buying multiple bundles, but Scandinavia can be finished for free with substantial time left. These events last for 148 hours and have 24 ranks.
Long events include Halloween, Christmas, and Anniversary. These events last for 244 or 268 hours and have 30 ranks. All events in this category are nearly impossible to complete. To complete long events, you need to buy multiple bundles with real money and spend at least 800 gems. The amount of spending is far more than what the rewards are worth, so I don't recommend it, unless you have another reason for completing them besides rewards. Unlike normal and short events, these are not part of the normal rotation. If you do want to complete them, the Rank 10, 16, and 24 bundles are your best option.