You'll pick up this quest in Chapter 3, after clearing out Scrap Boulevard with Tifa. To pick up this quest you'll need to talk to a little girl named Betty, who can be found at what passes for the school in the Sector 7 Slums. Go get there head east, south and east from the Weapon Shop. Talk to Betty and she'll ask for help finding her friends, although her description sounds... kind of odd. Cats? Okay.
So, this is a bit of a scavenger hunt. Fortunately Tifa does a fairly good job at calling out these felines when you get near one. You can also ask Betty for hints. But this is a guide, so you probably just want to know where they are. Their locations will be given below, in text, but you can also use the provided map (see above) - their locations on the map are, in order, the right-most one, the middle one, then the left-most one.
To find the first cat location, head east from Betty and when you reach a fork continue northeast to reach a three-way intersection, where you'll find the cat. Interact with it and it'll run off.
For the second cat location you just need to head to Seventh Heaven, where you'll find the cat on the porch near the door. Being a cat, it's a pain in the ass and runs off again.
Third time's the charm? From Seventh Heaven head west past the Item Shop, and at the intersection just beyond said Item Shop not the two paths continuing west. If you take the northern path you'll just end up back at the Stargazer Heights apartment building, which is not where you want to go. Instead, head west along the southern path to find a woman standing near the shed you (may have) crawled through earlier en route to acquire two doses of Adrenaline. Talk to her and she'll mention a cat - a pastry thief, at that! Climb through the shed and you'll find your white cat amidst a congregation of other pests. After Cloud tries - and fails - a third time he'll give up.
Go report to Betty and after a moment of doubt the cats will show up. All's well that ends well, right? Anyways, you'll get a Maiden's Kiss for your reward - a consumable item that "removes toad-related curses". Might come in handy sometime later down the line, although leveling up that Cleansing Materia until you unlock Esuna is probably a better long-term strategy, but we digress...