It depends on the venue. Street handling is obviously going to differ from tack or autocross. Having just done almost 2 months of shopping for cars, I've got recent (though short) impressions of a lot of semi-sporty (definitely not 'sports', though) cars.
A good handling street car is reponsive but predictable. Excessive response makes the car twitchy - I like crisp turn-in, but I'll give up a little in order to have confidence in the car's repeatability. Some hotshoes like a lot of toe out in front because the car turns in instantly, but it may zig too much or too little. I try to mimic the Jackie Stewart "tennis ball" driving technique, so I prefer good linear response rather than ragged-edge jumpiness.
It's not exactly handling, per se, but I like responsive brakes. The WRX we drove had good brakes - they stopped the car every time, with confidence - but they felt like they had excessive pedal travel, which provoked a little subconcious "yikes!" every time you approached a turn at speed.
Damping is critical, and again, I will give up a touch of twitchiness in order to buy a little composure. So far the TSX seems to be very well damped - it never hops over irregularities, but it manages to stay fairly flat when cornering.
Obviously I pefer mild oversteer, but I'll deal with moderate understeer so long as it responds to throttle/brake. My ACR power understeers, but a quick lift tucks it right in, and you can provoke oversteer on demand. It's pretty balanced for an FWD car.
Probably the best-handling car I've ever driven is a Lamborghini Espada or a Maserati Bora. The Lambo's controls felt like a truck at moderate speed, but wailing down the two-lane blacktop upwards of 120, it was perfectly weighted. It had great road feeling, too.
[edit]
Actually, scratch that. The best-handling car I've driven is a Porsche 928. Within the limits of my driving skill, it handled every bit as well as the Espada, but it was much easier to drive at all speeds.