The rules span over 3 days, so not everyone in the whole city moves their cars all at once. First day, no parking on emergency routes, all other streets are fair game. Second day, park on either side of snow emergency routes, or on odd side of non-emergency routes. Third day, park on only even side of non-emergency routes, or anywhere on the emergency routes.
There are specific times for when you need to move your car each day, but if after 10am on day 1, they fully plow the emergency route to the curb, you can move your car there.
You can read about it here: http://www.ci.minneapolis.mn.us/snow/parking-info.asp
They announce "snow emergency is in effect" on the radio, tv, newspaper. You can't really miss it unless you live in a cave.