In my state, they (along with the Emergency Broadcast System) were part of the civil defense network created during the Cold War. They were initially intended for war, imminent threat of war, or other national crises. Their brief was later expanded to be used for state and local emergencies. Exactly what they’re used for (tornadoes, tsunamis, flood warning, etc) and under what conditions they can be used, depends on your state and local officials.

I’d also add that the government has tried to be rather restrictive about what conditions it could be used under. They want the siren use to be unusual enough that people respond to it as it there’s an actual emergency. Unlike, for example, the fire drills you’d have in school or work, where people would debate whether to bother leaving their desks, head back to the office to get their things, etc.