International flights to Iceland arrive in to Keflavik International airport (often referred to as Reykjavik).
Flights are available between the UK and Iceland with the following airlines:
Icelandair from Glasgow, London Gatwick, London Heathrow, Manchester (and from various European and US cities)
Easy jet from Birmingham, Bristol, Edinburgh, London Luton, London Stansted, Manchester
WOW Air from Gatwick, Edinburgh (and from various European cities)
SAS fly from Copenhagen to Iceland.