Mystery Gift is a gift starting in Gold/Silver. If using this you may get one of many gifts per day. While Gold/Silver are are not unsimilar there is one. You will get different mystery gifts. You need Wi-Fi Connection for it.
Starting with Generation IV, this is also used as the name of a program where one can download event Pokémon off of Wi-Fi or in-store events. In DPPt, one must unlock it by talking to a man in the Jubilife TV Station, then tell him "EVERYONE HAPPY, WI-FI CONNECTION". In later games, it was already included without having to unlock it.