Not much to add after the very good summary of reportyourlevel.
Perhaps mentioning that while it is not mandatory, a very good practice is for the atco starting the shift to arrive 2-3 minutes earlier and plug himself to the console to listen to the communications while checking the screen in order to get a feeling of the situation prior to the formal handover. Also very advisable is that the atco ending his shift stays put next to the console for another 2-3 minutes in order to be able to point out things that might have slipped the newcomers attention and to be able to give a hand if necessary.
Sadly this routine is not always used