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To expand on what avoid3d said:

You can make a plot of entropy vs. energy of a system. Further, you can calculate the derivative (slope) of that plot and give it a name. You could also calculate the quantity 1/slope and give that a name, too--and the name we give it is "temperature." So, the temperature of the system is negative any time the slope of entropy vs. energy is negative.

How do we get a negative slope? We need to find a place where entropy decreases when energy is added to the system. Entropy is proportional to the natural log of something called the "partition function (see below)." That means we need to find a place where partition function has a negative slope.

The partition function tells us essentially "how many distinct ways can we arrange (store) a given amount of energy in this system?" For almost all macroscopic systems, there are a greater number of ways to rearrange the system each time a unit of energy is added. However, it is possible to construct a system where adding a unit of energy actually restricts the number of ways you can arrange the system. And that is the basis for negative temperature.