Read-only feature:

Unlock a character-exclusive dungeon

Read+Write feature:

Summon an upgradeable illusory hero to help during battle

Write the character's stats to the amiibo to power up their illusory version.

Source: Mon Amiibo

Read-only feature:

Summon illusory units to help during battle

Each individual figure or card summons different units, some much stronger than others.

This feature works with any amiibo (with the possible exception of those that work with its other features).

Source: Reddit user ibroussard / Thanks: memoryman3