amiibo are physical toys created by Nintendo. They are based off characters from their games.
What makes them special is that they make use of NFC (near field communication) which allows the toys to send and receive small amounts of data when placed near to a NFC reader. This data is then used within the games.
Using NFC the amiibo can be used within a compatible game. So far this can take two forms.
From launch amiibo will be compatible with the Wii U. Nintendo has promised support on the 3DS in 2015 with the use of an accessory. The will also be a new 3DS version called New 3DS released in the future which has amiibo support built in.
By using NFC (Near field communication). There is a tiny chip within the amiibo toy that can transmit and receive small amounts of data over very small distances.
It the same sort of technology that can be found in phones and bank cards.
Batteries are not required.
An amiibo can only store the data from one game at a time. But an amiibo can still be used to unlock items in other games.
At the moment you will need to use the amiibo setting menu found within the Wii U system settings menu.
It really depends how you look at it. In one way, features and items are locked away unless you have an amiibo. But these features are not needed to play the actual game, nor are they a major component of a game in the first place. They are, so far, more like an extra for having an amiibo figure. Or looked at another way you are getting a nice figure that as a bonus can interact with Nintendo games in fun ways.