Shannon Airport is about a 70 minute drive directly south of Galway. It is a much quieter airport than Dublin and is served by daily connections to the east coast of the United States, to several UK airports including London Heathrow, and to some other locations in Europe, mostly by Ryanair. There are hourly bus connections with Bus Éireann to Galway, taking 70 or 110 minutes depending on the number of stops (Route 51 has many stops along the way, but Route X51 has only one drop-off on the edge of Galway and one at the station in the city centre). If you are travelling from somewhere with a connection to Shannon, it is a good option for getting to Galway. Travellers to the United States clear US Customs and Immigration in Shannon and go through domestic arrivals in the US.
Ireland West Airport is about the same distance north of Galway but has fewer flight connections and is also less well connected to Galway by bus, although there are some services. It does have Ryanair flights to some destinations in Europe. Flight timetables are seasonal and Skyscanner is a good place to look.