The term streetable is used within automotive circles to describe a car that has been modified, but is still usable as a daily driver vehicle without too much effort, discomfort, or risk. The term comes from the 'ability' of the car to be driven legally, safely, and comfortably on a public street. It's a fairly subjective term, and many sports cars come from the factory in a state that most would consider totally unstreetable.
The main factors that affect streetability in a car are the positioning of the powerband, and the drivetrain setup. Also important are the suspension setup and ride height, and I'll be discussing all of these in that order.
In most cars, peak horsepower increases usually come at the cost of both narrowing the powerband, and moving it further up in the engine's rev range. This means the car has to be driven harder to get useful power out of it, which obviously impedes streetability. In the case of turbocharged cars running a lot of boost, another problem can be the suddenness of the turbo kick when the turbo spools up enough to start producing power. In some cases this can mean that if you're accelerating through a corner and the turbo starts doing some work, the sudden burst of exra power can be enough to cause the unwary to slide off the road and into the scenery.
If you modify a car's engine to produce significantly more power than the drivetrain was designed for, you'll need to upgrade certain components, first the clutch, then the gearbox, and then the differential - depending on what kind of increase we're talking about.
A tougher clutch can tend to be more 'grabby' and sudden in its take up. Uprated clutches also tend to have a stiffer pedal action, and these factors combine to make a car much harder to drive in traffic, where you're starting and stopping all the time.
Upgraded gearboxes are less problematic, and simply using stronger than standard components shouldn't make the car any harder to drive, but using a straight cut (dog) gearbox (As many choose to, in order to lose less power to the gearbox) can make changing gears without clashing them a real challenge, as anyone who has driven a car with worn out synchromesh rings can testify.
An upgraded differential should also not really be a problem unless you go so far as to use a locked diff, or a VERY tight LSD setup, in which case turning corners will fry your tyres every time, and the car will be very disinclined to turn in. This is expensive, and is a nuisance.
It may look cool to have your car slammed down to within an inch of the ground, but if you can't get over speed bumps, or you regularly have to drive on rough roads, you'll get sick of it pretty quickly. Most standard passenger cars can stand to be lowered a little and have slightly stiffer springs fitted, but go too far and you'll end up bouncing all over the road and beaching yourself on speed bumps.