"Genève Aéroport" is a relatively small airport very close to Geneva and the French border. Switzerland and France share this airport. Geneva (Switzerland) is 5 km away, Lyon (France) is 150 km away. Most flights originate from countries in and around Europe, although there are some intercontinental destinations too. Most intercontinental flights arrive at Zurich airport.
Transportation from the airport to Geneva
The easiest way to reach Geneva and the main station is by train. It's a 7-minute ride. From the Geneva main station, you can take trains to France and further into Switzerland. The rail station can be found at the lowest level of the airport.
You can take a bus from the airport to Geneva as well. If you have a lot of luggage you may want to book a private or shared transfer.
Rental cars are available at the Arrival level of the airport. If you continue into France you can get a rental car in the French sector of the Check-in level.
A cheap "Unireso zone-10" ticket is available to get to the Geneva main train station. It costs CHF 3 and is valid for 1 hour. Note that you don't need a ticket if you have a rail pass that allows free traveling throughout Switzerland, such as the Swiss Travel Pass.
Free public transport in Geneva
Hotel guests in Geneva can travel by public transport for free. More information can be found here.
The Swiss Travel Pass and several other rail passes also cover the train and other public transport in Geneva for free.
Hotels near Geneva airport
There are several hotels very close to the airport. They're a good option for a late arrival or early departure, especially since some of them offer a free shuttle bus service to and from the airport.
Geneva is really close to the airport, so you can easily stay in one of the hotels in town as well.