Amiibo are figurines of Smash Bros. characters that are sold separately. They use near-field communication to interact with New 3DS. Amiibo are currently only compatible with the New 3DS (and Wii U) but will work with regular 3DS systems in the future, with the release of a amiibo Portal accessory. The same amiibo can be used on both the 3DS and Wii U versions of the game.
Select the amiibo option from Games & More to get started. This option only appears if you're playing on the New 3DS. Simply touch the amiibo to the bottom screen and hold it there for a moment. You'll be prompted to give your amiibo a name and set its costume. Don't think about it too much. You can change both of these things later.
You can set your amiibo's special moves, choosing from any default and custom specials you've unlocked. You can also feed equipment pieces to your amiibo (including character-specific equipment they can't normally equip) to raise and lower their stats and learn abilities. You can only feed your amiibo a few pieces of equipment before it becomes "full" and will need to fight to build up its appetite again. Your amiibo can only know three abilities at a time, but which one (if any) it forgets when fed a new piece of equipment holding a bonus effect is up to you. Stat changes are permanent, so don't feed your amiibo anything if you want it to keep its default stats for the sake of keeping things fair.
Amiibo can be used in Smash modes, either from the amiibo menu or by pressing ZR and scanning the figurine at the character select screen from the regular Smash menu. As an amiibo fights, it will learn and level up, becoming smarter. It will gain more EXP if you defeat it and if it fights a variety of different opponents. The maximum level is 50.
The amiibo's fighting style will change depending on how you approach it. For example, my Shulk amiibo likes to use Hyper Monado Arts (Buster) and land a few hits to build up damage before switching to Hyper Monado Arts (Smash) to finish the opponent off. It is also good at landing its down special counter move, because it has been countered the same way many times. Amiibo fighters will continue to learn even after they've hit level 50.
As it fights, your amiibo will find prizes, such as piece of equipment and trophies. Scan the amiibo in the amiibo menu to claim these goodies for yourself. When you've finished battling with your amiibo, you'll be prompted to tweak it. Makes sure you do this and save your changes before shutting off your system, to save your progress to your amiibo figurine by scanning it again.