An airport in the most simple of terms is simply the location where airplanes (and other aircraft) take off and land. Most airports, however, include many other functions.
The Landing Strip
The landing strip is the most vital part of an airport. It is a flat hard surface that is long enough for an airplane to gain enough speed to take off and slow down for landing. If the planes can't do either of these two things the rest of the airport is unnecessary.
Maintenance and Refueling Areas
Most modern airports have areas for plane maintenance and refueling. Planes need fuel to get to their next stop and regular maintenance for safe travel. Fuel is heavy and it would certainly be inefficient (and unsafe) for a plane to carry the fuel required to get to its next stop.
Passenger and Cargo Areas
If a plane is going to carry people or goods it needs an efficient system of loading and unloading them. There are many companies that offer air travel services and one airport can have many different carriers coming and going.
People use a ticket system where they purchase a ticket and at a later point in time (hours, minutes, or days) the exchange the ticket for a ride to their destination. This frequently includes standing in various lines, security checkpoints, entering, and finally exiting the plane at the destination of choice.
Air Traffic Control
In order to safely have many planes take off from an airport in a short span of time a method of controlling the coming and going of the planes has to be enacted. Coordinating these efforts is done through the air traffic control systems. This determines which plane is taking off, which plane is landing, and hopefully no one crashes into anyone else.
The larger the city that the airport is located in will normally determine how complex the system is to use it. A small town that just has a few private plane owners using it will require much less coordination than a large international airport which serves many thousands of people per day.
The regulations that the airport is required to follow is determined by the country in which it is located. Different countries have varying levels of concern for the safety and well being of the passengers and cargo being transported.
There are regulations for every aspect of air travel as it is a very commonplace mode of transportation that millions of people use every year.