Nintendo-themed figurines with NFC chips embedded that have various effects in various games, typically unlocking stages or costumes or characters.
Amiibo cards are the same chip, but instead of being in a plastic figure they are embedded into a card. The Animal Crossing ones are the most popular, with a card for each character in the game. In the games they work with (Happy Home Designer, New Leaf, New Horizons) they let you move that character into your town, get items or DIY recipes from them or design their homes.
There are also series for other games. IIRC one of those Mario-themed football games (I guess the WiiU one), and a Power Pro game.
Just keeping track of games relevant to me that do stuff with Amiibo.
Many characters have multiple Amiibo, and usually they will all unlock the same thing.
Each Fire Emblem Amiibo unlocks a piece of music for the Auxiliary Battles. Scanning a single FE Amiibo is supposed to unlock an extra location in the second half of the game, and an extra ability in New Game Plus.
Scanning any Amiibo will make items spawn around the Amiibo Gazebo.
Each Amiibo unlocks a weapon for that character, and the Rabbid version of that character.
Unlock one stage each.