TIme-Limited Events are races that are set at a location, and are slightly based on multiplayer. While it does not use multiplayer as its game mode, you can see some races that are multiplayer-only events (usually Multiplayer Weekend Cups as well as Assembly Cups). These events require Fuel, that replenish anytime the fuel tank is not full (10 slots).
Every day, there is an event that is named "The (day) Heat", that all can enter, and play against MAXed cars (usually mid-high to higher-end cars of the class) at one location for one lap. You can pick any car you have in the inventory. This event is the least-sourceful for obtaining blueprints, but sometimes some pretty good rewards come out of getting a good lap time. After you complete a race, you'll be sorted into a position on the leaderboard, and depending on that spot, you can earn rewards. These races vary with the location, and the class that is only allowed in that race also gets switched up every day. These races start at 8:00pm the day before (If the event were to start on a Tuesday, it would start Monday at 8.), and end at 8:00pm the next day. These events usually require two to three (more common) Fuel slots.
As of the Blueprints update, various assembly cups have opened. These events are five days in length, and can give you blueprints for a certain car. In the event, you will get the car at MAX/PRO level for FREE (even if you don't have it), and you will be entered into a race. These events could be timed, or they could be multiplayer matches. If it is a multiplayer match, you will earn points at the end of the race, and those points are what the leaderboard needs in order to position you. Depending on the spot you finish, the amount of points will vary. The only negative thing about multiplayer is that you commonly get racers that are much higher in points, and really just beat the crap out of you, causing you to go into a rampage and take your device and throw it across the room like some little kid who doesn't like losing (it's okay, I do it too.), and when you enter the event, you can only see how many racers are joining in, and you can't see their names or their scorecard like in the actual Multiplayer mode. These events usually require three slots of Fuel.
Occasionally, you'll get one to two day cups for a special kind of day. If there is a National day for something (Like Back to School, Valentine's Day, or March Madness), there's a cup for that. This gives you the chance to try MAX/PRO beasts that can't be obtainable from the shop without competing in an R&D and winning it (in order to get the car), as well as earning some pretty big rewards. This event doesn't require the car to be in your inventory, and it doesn't require it to be MAX/PRO to enter, as it will automatically give you the car fully upgraded. This event usually costs two to three Fuel slots.
Location cups are centered around ONE location, and require a certain group of car within a select range. These can all be in the same class, or it can jump between classes. This event can sometimes cost one to three Fuel slots.
Apex Wednesday cups are similar to Daily Heats, but they are focused on one prize, which can vary from boosters, tokens, or decals. These events start Tuesday evenings at 8, and end at 8 on Wednesday evening.
Rarely does an event come up that is seven (yes, SEVEN) days in length, and there's only one event that can hold that long of a tag. Trophy races. These events use for the most part one car, and require you to go through a total of SEVEN courses (for each day). Based on the trophy requirements on the left of the race thumbnail, there is three different qualifications for that course that you have to complete. Those trophies can give you exclusive rewards. Some of these events require the car or cars, and some give it to you to try for free (Like the Simple Mobile Cup with the Arash AF10). These events are usually the hardest to complete, as you can easily miss a race for the day, and lock yourself out from getting the top prize. These events require three Fuel slots in order to enter the race.
After the event ends, and you've participated in the event, you will need to wait a little before claiming your rewards, as the finalizing the leaderboard continues. If a new event started, you will see a small, overlapping icon colored in green with a clock on it. If your rewards are ready and you need to claim them, there will be a yellow overlapping icon with an ! on it. If you have finished an event, and don't claim it within two weeks, you will no longer be able to claim your rewards. Whether or not this is true as of the most current updates, is something to be figured out. Best collect them when you get the chance.