The concept of measuring time is a very complicated one, but without getting too in depth, our concept of time comes mostly from the position of the sun in the sky at different moments in the day or the year.
The concept of a day comes from the fact that the Earth has a light and dark side. The light side is the half of the Earth that is facing the Sun, the dark side is the half facing away. As the Earth rotates, you pass from dark side, to light side, and then back to dark side over and over again. This is what makes your day. The South Pole is a tricky place because it is located on one end of the Earth's rotational axis, the imaginary rod that the Earth is spinning on. When you are on the axis of rotation, you spin in one place, as opposed to moving in a circle. This is like being on a merry-go-round, the axis is in the center and the animals you sit on are on the edge. Since it is an the axis, the South Pole doesnt ever go through the normal process of passing from dark side to light.
The South Pole "day" is due to the inclination of the Earth's axis with respect to the plane of its orbit. What that means is that at any time of the year, there is one pole that is facing towards the Sun and one pole that is facing away. As the Earth moves around the Sun in its orbit, these poles switch roles, passing from light to dark. Thus, the Polar "day" is really one year long, with 6 months of sunlight and 6 months of darkness!!! I am going to the South Pole during the Polar Summer, or during their "daytime".