If we think that temperature is a measure of how much the atoms move around then it is not possible to have less than zero movement.
But if we think temperature as related to the ratio of how much a system's entropy changes when a certain amount of energy is added (or removed) to the system [1],
1/T = dS/dq
then it's possible to construct systems with a negative T.
[1] Eqs. 8–10 in http://en.wikipedia.org/wiki/Temperature#Second_law_of_therm...
It's just a matter of convention, a negative temperature just means that the temperature gets closer to 0 as you add energy. Did I get that correctly?
Well, kind of, but I think that would be an oversimplification.
Negative temperatures have a concrete physical meaning, being "hotter than infinity", in the sense that if we bring in contact 2 systems: one with an arbitrarily high positive temperature, and one with a negative temperature, energy (heat) will flow from the negative system to the positive.
This looks tidier if we use the thermodynamic beta (beta = 1/T) instead of temperature. Then we can say that heat always flows from a system with a smaller beta to systems with a larger beta.
http://en.wikipedia.org/wiki/Thermodynamic_beta
Maybe debt is a somewhat useful metaphor here. If someone has a debt of 5 apples, in some ways if makes sense to say that this person owns -5 apples. But in some other ways it makes no sense: a person with 5 apples can eat them to get less hungry. A person with -5 apples cannot eat them to get more hungry.
A subsequent publication[0][1] argues that their notion of "negative temperatures" is invalid as they used a flawed definition of entropy.
[0] http://scholar.google.com.au/scholar?cites=41283096248189415...
[1] http://www.physik.uni-augsburg.de/theo1/hanggi/Dunkel_Nature...