Cat's is by far the most architecturally beautiful hostel I've ever stayed in: the enclosed central courtyard of the building is not only a chilled meeting and relaxation point; it's also a gorgeous work of painted Spanish arches and columns, exquisite marble flooring, and an ornate stained-glass ceiling. As well as the building, the hostel itself is also a great place: it has a lively bar with cheap drinks (including sangria); a great mix of fellow guests; and many conveniences nearby. My only complaint is with some of the staff. Other than that, this is quite likely the best hostel to be found in Madrid.
Rooms: fairly quiet, and very secure (electronic key-operated doors and lockers — although some of the doors are hard to open). Most of the rooms also open onto the lovely courtyard.
People: an international bunch, but most of them are the backpacker crowd that we're all looking for.
Staff: a bit inefficient (particularly at checking people in); and I also had a bad experience personally, with one particular staff member. But they're generally friendly, and they're available 24/7.
Location: walking distance from Atocha central train station, from numerous good eating and nightlife places, and from Madrid's central square (Plaza del Sol).
Food: free breakfast is good, if a bit small (1 cereal or croissant, 1 tea or coffee, 1 juice).
Internet: 4 PCs in the downstairs bar, available for free for 20 minutes (they're very popular — you'll have to wait to use them). Alternatively, numerous €1/hr places can be found nearby.
Hot water: pretty good, although they have the usual annoying "water saving" whack-and-whack-again taps. There are also plenty of bathrooms available.