There is no airport that directly serves Cinque Terre, so you will have to fly into a nearby airport in order to get there.
The closest airports to Cinque Terre are Pisa airport (PSA) and Genoa airport (GOA). The Milan airport and Florence airport are not far either, and many intercontinental travellers will land there when they arrive in Europe.
That said, Pisa airport and Genoa airport are closer and more convenient for getting to Cinque Terre, so try to fly into one or the other if you can.
To get from Pisa airport or Genoa airport to Cinque Terre, the best option is to take the train. To find out more about how to do this, you can take a look at the Pisa Airport to Cinque Terre or Genoa Airport to Cinque Terre guide for details.