Storms (a.k.a. typhoons) are events that only occur during the summer season.
Storms destroy fences, have a chance of wiping out crops that you've planted, regenerates weeds, and can sweep branches and stones on your property. They are highly destructive, and can hinder some of the progress you've made in game. It can also have a negative effect on any animals left outside in a storm. The animal may become sick, and will be very unhappy. In most games, you will be unable to leave your house when a storm is taking place, and your only option is to sleep, wasting an entire day. In other games, while you are allowed to leave your house, your character will have a high risk of getting a cold, and your stamina will drain drastically when doing tasks.