First of all, amiibo are basically like Skylanders and Disney Infinity figures in a sense, But when Nintendo makes a product based on old ideas they always improve it. The original line of amiibo figures is and was the SSB (Super Smash Bros). And Super Smash bros was the very first game franchise to use the concept of toys coming to life on your TV screen (as far as I know). Since the original SSB on N64, Gamers have wanted to own those Nintendo figures. SSB Melee for GCN (Game Cube) was the first SSB game to put the figures on a form of base which transforms and gives us the playable characters. Anyways, moving onward... amiibo are NFC figures, NFC stands for "Near Field Communication"
The same thing that some Android devices have in them is in amiibo. The GamePad for the WiiU has a little spot down on the bottom left side where you can scan a figure, this spot is called "The NFC Scanner." Some games Nintendo releases use amiibo for multiple purposes, but not ALL amiibo work on ALL games, there are games like SSB where you can use your SSB amiibo as a fighter and as a teammate. And you can use those same amiibo in MK8 (Mario Kart 8) to unlock costumes for your Mii character to race in. In short: you can use amiibo to unlock bonus content or to use in games for the WiiU and some games for 3DS as well. The old 3DS/3DSXL consoles don't have an NFC scanner like the WiiU GamePad has. Because they don't have one built in, Nintendo's releasing an accessory/adapter to use amiibo and soon-to-be-made amiibo cards on the 3DS consoles. Otherwise, you can buy a New 3DS console today and use the built in NFC Scanner in the lower screen by simply tapping the amiibo to it. When you tap an amiibo to your scanner, you can give it a nickname (my SSB Mario is "Red Wonder") and assign to it an official Mii Character/owner. You can do this in any game the Amiibo is supported in and I believe even through the console's/system's HOME menu. This is my personal definition of the amiibo figures:
The amiibo figures are an optional accessory for WiiU/3DS consoles, they're also an optional enhancement to games which work with them, they'e inexpensive and an improvement on the figures-to-life genre. They took an idea and applied an older concept to make it what Nintendo fans have always wanted, the option to play with real Nintendo figures in their games.
What I've said throughout this article is that Nintendo improved on the Toys-to-life genre. What I mean't is that Nintendo say the Skylanders and Disney infinity figures and decided to apply their old concept of smash bros trophies/figures coming to life and fighting. Also, Skylanders have to stay on their portal the entire time to be used in their games, while amiibo only have to be tapped to be used.
My opinion on Nintendo's lack of quantity for amiibo releases is: Nintendo isn't truly responsible for the amiibo figures lack of availability, I've been informed that Nintendo practically asks how many a store wants. And the stores choose how many they get. So if you don't get the amiibo you wan't, don't take the complaint to Nintendo, take it to the store. Nintendo has even formally apologized before to the public for not fully supplying the demand, they said they didn't even realize how popular the amiibo would be.
As to why they named their NFC figures amiibo, I have a theory. The term/word amigo definition from Google is: "used to address or refer to a friend, chiefly in Spanish-speaking areas." Nintendo may have thought of this word when planning on making the original SSB lineup of amiibo, for instance you take the word amigo then add one letter i and replace the g with the letter b, and you have amiibo. : amigo...amiigo...amiibo. See how that works? This is simply a theory of mine, but in my opinion its a possibility.
Source: Nintendo's site/What is an amiibo figure?
Game on and Keep it Rehl!.