The taxis are very cheap but you should pay attention to a few important details. The fare, the company name and the car ID must be clearly displayed on the door of the car. Once you get in the car, the ID of the driver has to be clearly displayed and the meter must work. The standard fare is 1.39 RON and most local taxi companies use it. The fare for a drive from any point in Bucharest to the city center should not cost more than 20-25 lei (4-5 euro). There is no night fare or airport fare. The same fare applies no matter the conditions. You can find a list with a few official taxi companies here. The most popular are Meridian, Cobalcescu, Speed Taxi, Apolodor, Leone, Cristaxi. You can also use smartphone apps to call the cabs, see StarTaxi – but you need 3G connection.
If you are coming from the airport, use the existing taxi ticket counters in the waiting area, after you have collected your luggage. Pay attention to the fare. See what others are doing and ask for help if unsure.
Note that, as in all the crowded cities, most of the taxi drivers are crazy and drive like there is no tomorrow. Do not panic :). You can just ask them to slow down (they will understand).
Of course, there is a minority of private taxi drivers that plays dirty tricks. They usually wait in very popular places (such as the Old City). As a rule, never go with a driver that does not start the meter. Never go with a driver that tries to negotiate (unless you know what you’re doing). Never go with a car that does not have the fare written on the door (pay attention on how it is written: E.g. Taxi Legal3,9 lei is one of the tricks they use – that’s a 3,9 lei fare). If anything looks suspicious, just ask him to stop and get out of the car. These being said, the worst that might happen is that you pay 3-4 times more.