A hit is super effective when the attacking move's type beats the defending Pokémon's type, dealing at least double damage, or quadruple damage against a dual type Pokémon weak on both types. The games announce it with the message that the move is super effective.
Super effective is the in game label for the 2 times and 4 times steps of the type chart. One weakness gives double damage. Two weaknesses, which only happen against a dual type defender, give quadruple damage, the hardest standard hit in the game. The opposite labels exist too: a resisted hit is announced as not very effective at half or quarter damage, and an immune hit is announced as having no effect.
The label matters because it confirms your read mid battle. If you predicted a switch and hit the incoming Pokémon super effectively, the message tells you the prediction landed before the damage number does. Players also use it backwards: an unexpected not very effective message against a Pokémon you thought you knew usually means a type change, an Ability such as Wonder Guard or Levitate, or in Generation IX a Terastallization you missed.
Damage still decides the knockout, not the label alone. A weak super effective move can fail to knock out a bulky target, and a strong neutral STAB move can out damage it, which is why experienced players run rough calculations instead of just chasing the message. Sounds help too, since the games play a sharper hit sound on super effective damage.
In play"That Ice Beam is super effective on both of Dragonite's types, so it goes down hard."