Weekly Events mainly effect two things: items in the Store, and PVP. Each Event is themed around a specific Kingdom. As one might suspect, these Events begin at the start of each week.
The Store has different purchasable bundles found in the "Weekly Event" and "Glory Reward" sections.
Event Chests also change their rewards with Weekly Events - their contents are based on the current Event's specific Kingdom, and will largely consist of troops from that Kingdom for the entire week.
PVP is changed in a few ways by Weekly Events, namely in that the Ranked PVP Leaderboard (see this article) is reset at the start of each Event, renewing the competition. All players' PVP Tiers are reset to the lowest level, meaning that they should play their way back to the highest tier in order to capitalize on the higher daily Glory Rewards.
There are also bonuses that last for the entire week - statistical increases for troops of a specific Kingdom (usually the one featured in the Event) as well as for troops of a specific type. Then there are conditional rewards for using certain troops or performing specific activities - such as increased gold or other resources from a PVP victory.