A Weekly Event is is an event that lasts a full week and then resets. The Arena is the only true weekly event, starting over every week.
Doggy, Giant Knight and Kraken events are bi-weekly on a rotation. One of the three events starts every other week in a specific order, starting with Kraken, then Giant Knight, and finally Doggy, before starting over again with Kraken. These events also last a full week once begun.
During these three Weekly Events, a special chest becomes available. This chest contains modules and heroes that are strong against the current event. For example: The Pirate Chest becomes available during the Giant Knight event because Explosives are strong against Navarre, and Explosives are the Pirates' specialty.