Read-only feature:

Receive a particular power-up depending on the character

Exactly which power-up you receive depends on the amiibo scanned:

Cat Mario: White Cat Mario Bell (invincibility)
Cat Peach: Random power-up
Other Mario characters: Super Star (temporary invincibility)

Bowser and Bowser Jr. only grant a Super Star in the main game. See below for their functionality in Bowser's Fury mode.

Read-only feature:

Unleash a powerful shockwave to knock out nearby enemies and blocks (in Bowser's Fury mode)

Read-only feature:

Receive a 1-Up Mushroom (main game) / Receive a Super Mushroom (Bowser's Fury mode)

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