Amiibos are a new Nintendo product. They are figures that you put on the Wii U Gamepad and it works with your game.
How to find if the game has is compatible with Amiibos
Games will have a symbol on it showing if it is compatible
Here is a list of games that Amiibos are compatible with or will be compatible with:
Super Smash Bros. For the Wii U
Super Smash Bros. For the 3DS
Mario Kart 8
Kirby and The Rainbow Curse
Mario Party 10
Captain Toad: Treasure Tracker
Code Name: S.T.E.A.M
Yoshi’s Woolly World
What Amiibos do
Some make it so it gives you items (Like Hyrule Warriors) but some do something else. The only one right now that does something else is Super Smash Bros. for the Wii U. The Smash Bros. for the Wii U lets you train you Amiibo. You can level it up to level 50 by fight it.
You can battle Ambiio by placing one on the square at the Select you character screen.
For More detail go to http://www.nintendo.com/amiibo/