Although I have known the "idea" of a parabola for quite a long time, my first formal introduction to the concept came at the Museum of Science and Industry in Chicago. An exhibit at the museum allowed you to adjust the angle of elevation, power, and other details on a machine which would launch a basketball across the museum depending on the numbers you plugged into the program. The goal was to get the ball to into the hoop on the other end, meaning you had to experiment with different combinations of numbers to find the angle and power that gave you the best parabola, to try and get the basketball in the hoop. As a little kid, I was so intrigued by this concept. After this experience, it was easy to realize that parabolas were everywhere, especially in sports (which is a relatively big part of my family). In soccer, when you kick the ball; in diving, when you take off of the board, and in many other sports, the concept of the quadratic function is vital.
Here are some pictures to give a visual: