Sets

A set is active while each of its slots holds a fitting piece. There are three sets, and one outfit can have all three at once. Coats and dye colors don't matter.

A render of four characters in full outfits: a gray catboy in blue cat-ear headphones, a ginger catgirl waving, a calico catgirl jumping and a Siamese catboy sitting — A render, not a screenshot. The ginger and the calico outfits make all three sets. The two in headphones have no Cat Ears on, so they only make Alley Cat

Full Neko

Pieces: Cat Ears, a Cat Tail, any collar (Bell Collar, Velvet Choker or Neck Ribbon) and paws (Paw Gloves or Paw Mittens).

Steps are quieter: the step sound plays at 35% volume.
The jump is about a third of a block higher.
A meow also sends up four hearts.
N turns on Cat Eyes: Night Vision for 10 seconds. The cooldown is on the keys page.
A Bakeneko steals nothing from you and gives a stolen item back for catnip.

Completing the set for the first time grants the advancement Full Neko.

House Cat

Pieces: Cat Ears and any collar. Full Neko includes both, so it always comes with House Cat.

Cats and ocelots stop running away from you.
Cats and ocelots within 10 blocks walk after you and stop 2.5 blocks away. A cat that was told to sit, or one on a lead, stays where it is.
While a tamed cat is within 16 blocks, your time without sleep is reset every second, so phantoms don't come. The in-game text says "a cat nearby", but a stray cat or an ocelot doesn't count.

Alley Cat

Pieces: a Cat Tail, Paw Leg Warmers and Paw Slippers.

A fall of up to 10 blocks does no damage. In vanilla the limit is 3.
A fall of 4 blocks or more ends on all fours: a short crouch, a soft thud and a puff of dust. Other players see it too. Gliding with elytra doesn't count.

Nine such landings without damage complete the advancement Nine Lives, see advancements.

With any set

Cat Food restores 8 hunger points. Without a set it restores 2.