There is one airport in Krakow. Find their website here. Its full name is "John Paul II International Airport Kraków–Balice", and is located in the village/area called "Balice". The Polish word for airport is "lotnisko".
If you get a taxi from the ones waiting outside the airport you will pay between 60-100zł to get to the city center. If you book a taxi from a cheaper company you will pay about 30-40zł. (see our Krakow Taxi page for more details). Uber will be around this price (possibly slightly more expensive). There is wifi at the airport, and using uber avoids having to call up a Polish taxi company on the phone. If you book a taxi that isn't the official airport taxi's you will generally need to wait outside the Hilton hotel, as that is the place where taxis seem to like to pickup from.
Check on Jakdojade.pl for bus times. You will need to get a one-way zone II ticket (see our public transport in Krakow page for details/help).
Look for bus routes 208 or 292.
As of late 2015 there is a train station right outside the airport, so you can now get a train direct into town. The Main Train station it will take you to is about 5 min walk from the main square.
There is also another nearby airport called Katowice Airport. It takes about 40-50 minutes by car/taxi to travel between Krakow and Katowice Airport. Katowice airport has lots of alternative routes (and often cheaper than flying from Krakow).
Read our famous and comprehensive guide to everything you need to know before vistiing Kraków!