A number of international airlines maintain regular flights to Iceland from Europe and North America. The local carriers are Icelandair and Wow Air, who fly to a range of destinations on both sides of the Atlantic.
The international airport in Iceland is at Keflavík, a 45-minute drive from the capital, Reykjavík. There are regular buses to the centre of town run by both Reykjavík Excursions and Grayline, as well as a service run by the Icelandic bus company. 
Towns outside of Reykjavik can be reached through domestic flights from the airport in Reykjavík. Flights are operated by Flugfélagið Íslands. Strætó, the Icelandic bus company, also provides buses to rural areas. Although it is advised that you plan well before you go as many of these services are infrequent and are susceptible to disruption outside of the winter months due to the Icelandic weather.