The name of the game is Six Cats Under. It can be played online, and it's rather quick to play.
Backstory: you're the ghost of a recently deceased crazy cat lady. Your job is to free your cats from your home, because now you can't feed them, and you don't want them to starve.
All cats have their own personalities and names. For example Baroness is grumpy, Fredrick only cares about food, Mr. Spock likes to scratch furniture, etc. (Source: click on the cat and the ghost will say it.) This is relevant for gameplay because you need to make her cats interact in a certain way to open the door.
I'll provide the solution of the puzzle in the comments, but please try the game before using it. Otherwise you'll lose all those nice tidbits of narration from the ghost.