As is becoming evident around the community, having Headmaster Kel'Thuzad on the board with 6 "cat" minions causes him to add a 0 mana 1/1 Mr. Bigglesworth to your hand. He cares that the minion is a cat specifically, not just any beast, which isn't a visible minion type. What follows is what I believe is the complete list of minions considered "cats" by the game.

I haven't actually tested these - don't even have KT myself. But in the game's card data there's an unnamed tag that cat minions seem to have in common, number 1584. All the clips I've seen and reports I've heard of his effect working so far use minions with this tag, so I've compiled them all here.

Classic:
Stranglethorn Tiger
Jungle Panther
Panther - Summoned by Power of the Wild.
Druid of the Claw (Cat)
Druid of the Claw (Hybrid)
Savannah Highmane

Pre-Standard:
Mech-Bear-Cat - Sure why not
King of Beasts
Metaltooth Leaper

Druid of the Flame (Firecat)
Druid of the Flame (Hybrid)

Druid of the Saber (Lion)
Druid of the Saber (Panther)
Druid of the Saber (Hybrid) - Is it a double cat? Cat squared?

Year of the Kraken:
Cat in a Hat - Summoned by Cat Trick.

Alleycat
Tabbycat

Year of the Mammoth:
Sabretooth Stalker

Astral Tiger
Crystal Lion

Year of the Raven:
Druid of the Scythe (Dire Panther)
Druid of the Scythe (Hybrid)
Night Prowler
Black Cat

Shirvallah, the Tiger
Spirit of the Tiger
Tiger - Summoned by Spirit of the Tiger.
Springpaw
Halazzi, the Lynx
Lynx - Given by Springpaw and Halazzi.
Interestingly enough, Spirit of the Tiger is considered a cat according to this tag, but Spirit of the Lynx is not. Might be an oversight.

Year of the Dragon:
Pharaoh Cat
Salhet's Pride

Escaped Manasaber

Year of the Phoenix:
Mok'Nathal Lion

Manafeeder Panthara
Twilight Runner
Mr. Bigglesworth - (Not in the gallery) The token given by KT himself has the tag like normal, but it could be a special case in practice for all I know.

Other:

These ones are the choice cards for the Druid's Choose One options. Same card but different name. Not sure they can actually enter play but if they did, they'd be cats I guess.
Cat Form
Firecat Form
Lion Form
Panther Form
Dire Panther Form

Shadowmaw Panther - (Not in the gallery) A treasure card from the Rumble Run adventure.
Battlestarved Lynx - (Not in the gallery) Another treasure card from the Rumble Run adventure.

Mr. Bigglesworth - The original Bigglesworth summoned as an Animal Companion in the Naxxramas encounter apparently counts too, if you ever feel inclined to reunite them.

Again, these are just minions that share an apparent "cat" tag with no official name. There's no way to see KT's actual logic and I haven't tested them all myself so there's a possibility the rule could be more elaborate, like "if you have 6 'cats' that are also beasts," or "if you have 6 'cats' that cost 4 or less..." That probably isn't the case but if you know of anything on this list that doesn't work or anything not on the list that does, it'd be interesting to know. And if you know of any poor cat minions that fail to get Kel'Thuzad's attention, I'm sure Blizzard would want to know about it!