An alt (sometimes called a sockpuppet) is an Internet term for "alternate account". In Miiverse's case, alts are created to use Miiverse while banned on another account. Alts have existed since the beginning of Miiverse due to its strict administration
How alts used to work
Before mid-2015, you could create alts even if you were banned on a system. If you create an account on your Wii U and it ends up with a 2 week ban or a permanent ban, you could create an alt on the same system to access Miiverse.
How alts currently work
During 2015, an update was released for Miiverse that prevented users from creating Miiverse alts if they had a ban on their system. Although this update hasn't removed alts entirely, it has drastically changed how alts work. Alts can be created if you have another system.
By hacking a system in the 3DS family and changing the serial number, you can bypass console bans.