Geneva's free transport card is handed out by your accommodation once you check in, and it covers trams, buses, the small lake boats and trains within the city zone, but not the ride in from the airport, which has its own free ticket.
At the reception desk of a mid-range Geneva hotel, a traveller is turning a small paper card over in their hands, looking for a barcode or an expiry date and finding neither. The receptionist explains, in the half-patient way of someone who has said this a hundred times, that the card is simply valid for the length of the stay and does not need to be scanned. Down the corridor another guest is already asking whether the same card will get them to the airport that evening, and being told, gently, that it will not.
Yes, Geneva's public transport card is free. It is handed to you by your accommodation, usually at check-in, and it covers unlimited travel on trams, buses and the small lake boats, plus trains within the city zone, for the length of your stay. It does not cover the specific trip from the airport into town, which uses a separate free ticket you get at the airport itself, nor does it work once you are outside the zone, for example on the way to Lausanne.
Hotels and hostels issue the card as a matter of course, usually with your room key or alongside it. Campsites do the same. The grey area is holiday flats: an Airbnb host is meant to provide the card too, since the scheme is funded through the city's visitor tax and applies to any paid overnight stay, but in practice this depends on the host knowing the rules and having a supply of cards to hand out. If your host has not mentioned it, ask directly when you arrive, and if none is offered, ask where the nearest tourist office or transport counter is, since they can often issue one on production of a booking confirmation.
The exact wording of who qualifies, and for how long the card runs, sits on Geneva's own transport page rather than in any traveller's memory.
Within the city, the card covers all the ordinary ways of getting about: the tram network, the city buses, and the small lake boats that cross between the two banks, which are a pleasant alternative to walking round by the Mont-Blanc bridge. It also covers trains, but only within the local zone that covers the city and its immediate surroundings, not journeys that leave that zone. Each of these is a separate confirmation worth having straight in your head before you set off, because the one that trips people up is the trains, which feel like they should be covered everywhere and are not.
Geneva Airport sits inside the same transport zone as the city, and there is a train link into the centre that takes only a few minutes, but the free Geneva transport card is not what gets you on it. Instead, there is a separate free ticket, issued from a machine in the arrivals hall before you reach baggage reclaim or just after, depending on the terminal layout. You take a paper ticket from this machine and it is valid for a single journey into the city on the train, tram or bus, for a set period after you print it. The Geneva Transport Card given to you at your hotel only starts working once you have checked in and received it, which is, of course, after you have already needed to get there.
The card's zone is generous for a short stay in Geneva itself, but it ends at a fairly tight boundary. Day trips that feel local, such as a run along the lake, generally step outside it. If your plans include anywhere further round the lake, treat the card as covering Geneva and nothing beyond, and buy a separate ticket for the rest of the journey.
Lausanne and Montreux each run their own version of the same idea, issued the same way, through your accommodation there, and covering their own local networks. They are not interchangeable with Geneva's card, and a card from one city will not be accepted as proof of entitlement in another; you need the one for wherever you are actually staying that night.
The mistake seen most often is assuming the card is a general-purpose rail pass for the whole region, and using it, or trying to use it, on a train to Lausanne. The ticket inspectors on Swiss trains do check, and a card that only entitles you to travel within the Geneva zone will not cover a longer journey. The fix is simple: treat the card as a city pass, check the coverage map at your hotel or on Geneva's own page before you plan a day trip, and buy a normal ticket for anything that leaves the zone.
Once the card itself is sorted, the next question is usually how to get from Geneva to wherever you are going next, whether that is Lausanne, Montreux or further afield, and what that onward ticket costs and where to buy it. SBB sells point-to-point tickets for all of these, and it is worth checking the fare and timetable the evening before rather than at the station counter on the day.