Pros: Tighter U-turns; good for heavy-duty towing and hauling; rear-wheel drive is arguably optimal for performance driving
Cons: Less traction on slippery roads; typically more expensive and less fuel efficient than front-wheel drive
If you're shopping for a car and you see the term "rear-wheel drive," it means that the car's driven wheels -- i.e., the wheels that receive power from the engine -- are the ones in back.
In a rear-wheel-drive (RWD) vehicle, the front wheels don't drive the car at all; their only job is to steer. This has a few advantages relative to a front-wheel-drive (FWD) layout.
First, because there are fewer components between the front wheels, those wheels are able to turn more sharply. All else being equal, that means you'll notice significantly tighter U-turns and such.
Second, whereas FWD vehicles are prone to torque steer -- i.e., sideways tugs at the steering wheel during hard acceleration -- RWD cars should accelerate perfectly straight every time, because the front wheels aren't burdened by power from the engine. So if you have a heavy foot or just appreciate consistency, an RWD car might be preferable.
Third, if the vehicle is tow-rated, RWD increases the maximum weight it can tow. That's partly because the driven wheels are closer to the trailer hitch, and partly because RWD vehicles tend to be more robustly constructed between those driven wheels. RWD payload limits (the maximum weight you can carry within the vehicle itself) tend to be superior, as well.
Finally, those who enjoy spirited back-road driving and/or weekend track days will generally get better and more entertaining performance from a RWD car, because the clean division of labor (steering in front, power in back) maximizes vehicle control for skilled pilots.
Drawbacks to RWD are minimal, especially with the advent of advanced stability control systems. However, no amount of technology can change the fact that a FWD car's engine is above the driven wheels, and this extra weight improves traction in slippery conditions. Aside from rear-engined oddballs like the Porsche 911 and Smart ForTwo, almost every RWD vehicle has its engine in front.
Also, RWD vehicles without stability control can spin out more easily in corners, which is particularly notable if you're looking at an older RWD car.
Moreover, RWD vehicles tend to be more expensive and less fuel efficient than FWD vehicles these days. There's nothing inherently costlier or thirstier about RWD, but FWD has become the norm for affordable mainstream vehicles, so that's just the way it is.
What it means to you: Modern RWD vehicles with stability control are not your grandfather's Oldsmobile -- they're really quite safe. But if your winters get chilly, FWD or all-wheel drive might be preferable.