The class of an airport is determined by the population size of the city it is in. The class and population size cannot be changed. Each city name has a colored dot next to it to indicate it's class. A class 1 airport will have a black dot and can only accept class one (smaller) planes. Class 2 airports have a blue dot, are larger and can accept both class 1 and 2 planes. Class 3 airports have a red dot and can accept all classes of planes. Higher class airports not only accept larger planes but they also have more layover spots, more passengers/cargo which want to travel through them and are more expensive to unlock.
Check out the Airpedia for all of the planes and their classes. You can also see an individual plane's class and other stats by clicking on the plane info button (icon with a plane and a magnifying glass). To start with, you are given class 1 planes. As you level up larger planes become available to you.