Nintendo can offer a lot to people who play couch co-op, but most of their games also include single-player or online multiplayer options.
(I've added a ☨ symbol to indicate which of the games have local multiplayer features. However, all games in the preceding paragraph are primarily single-player. I'm also leaving out several of the good Wii U games which are also available on PC, even though some of them have some pretty cool gamepad features. Rayman Legends is friggin' awesome on Wii U.)
Amiibos are basically tiny plastic statues with a small NFC chip inside. That chip interacts with the NFC reader built in to the Wii U gamepad to allow wireless communication in close proximity. Certain games will prompt the user to "connect" an amiibo for certain features. Very few games have implemented this functionality thus far.
Certain amiibos permanently unlock costumes for Mii Racers in MK8. Legend of Zelda-themed amiibos grant special items in Hyrule Warriors; all other amiibos grant generic bonuses. In Smash, each amiibo character corresponds with its respective Smash fighter - by connecting an amiibo to Smash, you can create a custom character that fights on its own. You can name it, choose its costume, select its moves, and equip/feed it with gear to give it bonuses. It will level up and learn to fight as its experience increases. It will learn from you in a crude way. It does not impart any bonuses to you in online mode. Basically, it's a robotic Smash buddy.