The airport in Reykjavík is the main hub for domestic flights in Iceland. There are schedueld flights form there to eleven destinations in Iceland and some in Greenland as well.
Icelandair flies from Reykjavík to Ísafjörður, Akureyri, Egilsstaðir and Vestmannaeyjar. Icelandair also goes to destinations in Greenland.
Eagle air flies from Reykjavík to Hornafjörður and Húsavík.
Norlandair flies from Reykjavík to Bíldudalur and Gjögur and from Akureyri to Grímsey, Vopnafjörður and Þórshöfn. Norlandair also goes to destinations in Greenland.
On the ISAVIA site you will find informatioin about all the airports in Iceland and departures and arrivals. ISAVIA also handles the operation and development of Keflavik International Airport in Iceland.
Icelandair is the main carrier flying to and from Iceland and serves many routes often in co-operaton with other airliners. For information on insternational flights by Icelandair, please visit threir website.
There are other airliners that fly to Iceland. How many at a given time of the year depends a lot. Here on the ISAVIA web site you can see which airlines are flying to and from Keflavík airport at a given time.