The lowest possible temperature when all molecules stop moving is 273.15 degress celcius. This is called Absolute Zero.
Actually, that should be minus 273.15 degrees Celsius (yes, it is capitalized). Otherwise known as 0 Kelvin (absolute zero). But all molecules don't necessarily stop moving at absolute zero. The Pauli principle requires that identical fermions (which includes many molecules) must occupy different energy states no matter how low the temperature. So there is some minimum amount of motion, called the zero point motion.
One of the "cooler" low temperature effects is superfluidity of liquid helium. This comes about because helium atoms are not fermions, but rather bosons, and they can all occupy the same energy state.
Bufungla's negative temperatures come about this way (and this is from a part of my brain that has had many years of inactivity, so be warned). N.B. I'm not absolutely sure the [logarithms] below belong where I put them. For qualitative purposes they don't matter: if the number gets bigger so does the logarithm. Thermodynamics mavens put them in so the value of the numbers stays manageable.
The statistical definition of temperature is that it is [the logarithm of] the rate at which the entropy of a system changes as you add heat. Entropy is roughly, [the logarithm of] how many ways you can arrange this system so that it has the same total energy. Heat is equivalent to the total energy of the system. This definition of temperature correlates pretty well with the "what does the thermometer say" definition at normal temperatures. It gets goofy for special systems at extremely low temperatures.
Such a goofy "system" is a collection of atoms (e.g., 100 of them). Because the energy level is so low, the only freedom the atoms have to choose a state of motion is the direction of spin. For example, the spin could be "up" (clockwise) or "down" (counterclockwise). The "spin-up" state has a slightly higher energy than the "spin-down" state. So the highest energy state you can achieve is the one where all the spins are pointing up.
(Of course the system has other energy states that are much higher and involve the atoms shooting around at various high rates of speed. But the idea is that there is such a gap between the spin-only states and the lowest atom-shooting-around state that you can't get between them without "huge" energy transfers.)
There is only one way you can arrange your atoms with all the spins pointing up. If you drop the energy by one cog, so that one atom has its spin pointing down, you have 100 ways you arrange your system--any of the 100 atoms could be the "down" one. So if you start in the next-to-highest state and add enough heat to get to the highest state, you decrease the entropy of the system. So the temperature is negative.