Negative temperatures show up in systems with limited numbers of states, which rules out things with normal translational or vibrational freedom. Sometimes it happens with spin states in a magnetic field. Negative temperatures are hotter than infinite temperature, with states with more energy being more likely, where at infinite temperature all states would be equally likely. If it had unlimited energy states, it would have infinite energy, which is impossible.

Zero absolute temperature means it's in the lowest energy state, so you can't get colder. It appears in the thermodynamics as the change of the number of states with energy going as 1/T, so you can't smoothly go below zero. Lower temperatures allow more large-scale coherence, such as superfluidty and superconductivity, so at many orders of magnitude lower temperature than previously observed, it's reasonable to think new pheonomena like this might arise.