Dart flights help to stabilize the dart in the air.
The very first flights of the wooden darts were real feather flights. When later the brass darts came on the market the flights changed first to paper flights.
The most popular flights now are out of different synthetic materials like nylon or polyester. They come in a lot of different shapes, colours and designs and usually are sold in sets of three.
The flights are folded when you buy them and it is very important to unfold them that the wings are all in a 90° angle. You should be careful with that because when they are too flat they really can impair the flight of your darts.
As the shafts flights are a cheap part of the equipment but they are not really durable. When flights are damaged too much they'll impair the flight of the dart as well. Probably you'll need to experiment with flights for some time to find out which suit your darts and your throwing style.
Most popular are standard flights, a lot of the Pro players use them as well. Generally one can say that as a beginner it's better to use not really small flights, because the smaller the flight the less is the stabilizing effect and you would need an almost flawless throw to be accurate with them.
Beside those common flights there are some flight systems on the market out of a stiffer material which can't be folded and will always stay in the right angle and so always give you the best aerodynamic support. One of the newest of those system is the fit flight system of the Japanese manufacturer Cosmo Darts.
In the Darts Performance Centre shop you can buy an experiment pack by which you can try a lot of different flight/shaft combinations. That is a good way to find your ideal set-up. A review you can find here: Flight and Stems Experiment Pack