Amiibo is new thing in Nintendo wii u. You tap an amiibo on the wii u gamepad and BOOM - in every game something happens. Amiibo figures are based on Nintendo characters.
In SUPER SMASH BROS if you tap your amiibo on gamepad, your amiibo character appears in the game. You can level up your amiibo and then the amiibo can fight in your team or against you (50 max level).
In MARIO PARTY 10, if you tap your amiibo on gamepad, you unlock your amiibo board. Every amiibo have a specific board according to their personality.
In MARIO KART 8, if you tap your amiibo on gamepad, you unlock costumes. If you tap Mario amiibo you unlock Mario costume. You can put these costumes on your mii`s and then you look like a BOSS yourself.
In Yoshi´s WOOLY WORLD if you tap your Yoshi amiiibo on gamepad, the little yarn Yoshi appear in the game. This amiibo is not made of plastic, its made of yarn. Your amiibo Yoshi is gonna help you in the game.
UPDATE: You can use every amiibo in diffrent seires. MARIO, Link, toad, inkling who cears and then you can be that yoshi what the amiibo looks like.
In captain Toad treasure tracker if you tap Toad amiibo on gamepad little pixel Toad appears. You go to stage and you need to find him. Its fun to find Toad.
In Hyrule warriors if you tap your Link (Toon Link) amiibo on gamepad you unlock spinner. Other amiibos giving money or weapons. But remember you can´t do this too many times in one day.
In Splatoon if you tap your amiibo on gamepad you recieve missions based of amiibo what did you tap in. By completing these missions you get special weapons and gear that you get only from these specific amiibos.
If you tap in Kirby, King Dedede or Meta Knight amiibo on gamepad you can do different things. With Kirby amiibo you can dash whenever you want. with Meta Knight amiibo you can dash through weaker enemies an with King Dedede amiibo you gain extra health bars.