Nice question. I'm guessing that you're wondering why there is a minimum possible temperature. If you're concerned about why zero Kelvin is the lowest possible temperature (a.k.a. absolute zero), it's simply because scientists have decided to make the Kelvin scale an absolute temperature scale - that means the lowest possible temperature, by definition, is zero Kelvin.
Now, on to the tougher question: Why is there a coldest possible temperature? In the past, people have wondered whether such a thing even exists. The first known indication that there was a lowest temperature was demonstrated in 1702, by French physicist Guillaume Amontons. He used an air thermometer which was, simply put, made up of a column containing a fixed mass of air supporting mercury liquid above it. When the temperature goes up, the air expands and pushes the mercury up. He suggested that you could make a temperature scale proportional to that volume. Thus, one could tell the temperature by the height of the air column.
What if the temperature were so cold that the height of the air column became zero? After marking the boiling and freezing point of water at his scale's equivalent of 100°C and 0°C (using modern units) respectively, he figured that the temperature where the air volume would be zero corresponds to about -240°C . Therefore it seemed that a minimum temperature would exist. Further measurements by Johann Lambert in 1779 improved the accuracy of the value to -270°C, close to the current accepted value of -273.16°C.
After the laws of thermodynamics were discovered, we have a clearer picture of what is going on. Temperature is an indication of the average amount of internal energy an object possesses. You can think of it classically as a measure of how vigorously the atoms and molecules in the object are moving around randomly. The higher the temperature, the faster the movement; and it is this movement of the molecules of air in the air column that supports the mercury in Amontons' air thermometer, as the molecules slam into the bottom of the mercury. Now imagine the particles in an object moving slower and slower, so that its temperature decreases. But this cannot go on forever. At one point, the particles have to stop moving, and this corresponds theoretically to the coldest possible temperature. You cannot go colder than that – this explains the existence of a minimum possible temperature.
The modern picture is a little different. As you cool any gas down, it condenses to a liquid and then typically to a solid. So the gas-based definition of temperature doesn't work for very cold things. Quantum mechanics describes the possible states of some system. There's a lowest energy state, and absolute zero temperature just means that the system is sitting in that lowest energy state. The temperature just gives the rule for how the probability for being in some state falls off at higher state energies. As the temperature goes up. the system spends more time in higher energy states. There's a lowest temperature because there's a lowest energy state. There's no highest temperature because there's no highest energy state.
As a side note, scientists have cooled atoms to amazingly low temperatures close to absolute zero. The current record is about 450pK (that's about half a billionth of a Kelvin) achieved by researchers at MIT in 2003.
Thanks for the question, and hope this helps!
Reference: Wikipedia's article on Absolute zero http://en.wikipedia.org/wiki/Absolute_zero
(published on 05/24/11)