Automatic transmission
There is basically little that can go wrong with an automatic transmission. It shifts its gears by itself and protects itself from misuse. Apart from the latest kind of automated gearboxes, these transmissions shift slower than one can do with a manual gearbox. This means that one cannot drive very aggressively (skidding tyres, etc.). Besides, in most cases, people who drive with an automated gearbox are easy going and therefore more careful drivers.
Although it is hard to estimate the quality, it is possible to test an automatic gearbox. So, it is important that the gearbox does not shift gears when you select a fixed gear. These fixed gears can be found next to the Drive (D) mode. Fixed gears are important when you find yourself in a situation where you want to brake using your engine. It is also recommended to take a look at the gearbox dipstick. The oil on this stick should be clear, not dark and cloudy. Secondly the oil should not have a burning smell, this is a sign of burned clutch linings and an expensive repair.
Finally you just have to check if the transmission works accordingly. So, does it shift up to a high gear when you drive a constant speed. When you give full throttle, is there a kick down? The transmission should shift one or two gears down and not shift up until it reaches a high rev count.
Do mind that an automatic gearbox does wear over time. This means that eventually the gearbox needs to be overhauled. It is hard to determine if this will be after 100.000 miles or way past the 200.000 marker. It is advisable to contact the brand dealer to find out what the factory directives are.
A word of advice. A compact car with an automatic gearbox is in general hard to find. So, in some cases buying a car in need of repairs can be interesting. Take the overhauling costs into account and compare similar offers with a gearbox that is in good condition.
Conclusion
You can find out by checking the gearbox what the general condition of a car is. When the transmission shows excessive wear you can bet that the way the previous owner has taken care of the car is very poorly. This is especially important when you are shopping for a recent car. When you find out that the gearbox is abused, you should ask yourself if this is the best car for you.