Istanbul Atatürk Airport
Named after the founder of modern Turkey, Istanbul Atatürk Airport is located 20 km west of the country's largest city Istanbul. As Istanbul is situated partly in Europe and partly in Asia, it's the only major airport in the world that can be said to serve two continents.
It is no longer possible to purchase a visa on entering Turkey. Visitors are required to purchase a visa in advance online at https://www.evisa.gov.tr/en/. (Do not use any other website, which will be unofficial and probably charge an excessive fee.)
IST is the main hub for the national carrier Turkish Airlines which is the world's 4th biggest carrier by number of destinations. Also Atlasjet, Freebird Airlines and Onur Air are based here.
Note that people are working on commission at the airport trying to make you use special shuttle buses for very high fees (30+ TL), so for people who wish to travel more economically the Metro/tram-combination is easy and fairly quick, and offers very good value.
From the airport, there are various options for getting into Istanbul: you can take a taxi (about 35-40 TL to Taksim. There are no night fare in Istanbul anymore. The price would be the same at midnight or midday.
About the same to Sultanahmet), the express bus service run by the local airport service called "Havataş"  which departs half-hourly between 4AM-midnight and costs 10 TL to Taksim and Aksaray and Kozyatağı), the public bus (line #96T) run by İETT costing 5 TL(3.5 with İstanbulKart), which has fewer departure times now, due to Havatas, which is also a municipality engaged bus service.
By light rail
Then, there is the metro (6:00 - midnight) (signposted "light rail" in the airport, when you get outside the baggage claim its about a 10 minute walk in the airport to the metro line. Just follow the signs), which will take you directly to the Otogar (bus station) or to numerous stops within Istanbul (Aksaray in the city centre is the last stop, transfer stations for tram heading for deeper into old city is available at Zeytinburnu and Aksaray). It costs 3 TL, by token (+an extra 3 TL when boarding the tram) and getting to Aksaray takes around 45 minutes. It is possible to be at your bus departing from Otogar within less than one hour after landing by taking the metro.
When entering the metro station, you need to buy a jeton (token) for 3 lira. Just hand the cashier 3 lira and he'll give you a token, or use the automatic dispenser (Jetonmatik), which accepts banknotes (5 TL, 10 TL) as well as coins. Use 'select' to choose the number of jetons and then push 'ok'. They don't accept credit card or foreign currency here. This will get you on the red metro line (towards Aksaray). From this line, if you are going to Sultanahmet, you can transfer at Zeytinburnu you will need a second token for the transfer. From Zeytinburnu, take the blue tram line T1, towards Kabataş which passes by: Sultanahmet, Eminönu and Tophane. The trip from the airport to Sultanahmet takes about 60 min. Also consider buying an Instabulkart (see Get Around below) which will pay for itself within a few trips on public transit.
Eat and Drink
Note that food and drinks at the airport may cost up to five times more than in the city proper, like in other international airports. If you are traveling on budget and plan to spend some time at the airport, it may be wise to bring your own meals from town instead of buying them there. If you come from the Metro, there is a supermarket in the tunnel leading to the elevators / stairs to the airport proper where you can do some last-minute shopping.
As at most airports, money exchange is more expensive than in the city. In Istanbul the airport money exchange offices accept limited currencies. It is recommended to bring US dollars or Euros for the best exchange rates.