Are there negative absolute temperatures? Sort of. Physicists have defined a negative absolute temperature, but it's a bit flakey, for these reasons:
Here's a bit more detail. Certain physical systems have small set of energy levels that they can be in. For example, a laser uses this principle. The atoms (or molecules) that produce the lasing effect can be in one of a number of energy states. Normally, only a small percentage of the atoms are in the highest energy states; many more are in the low energy states. Scientists have found equations that describe how many of the atoms are in which energy state. As you might imagine, these equations depend on temperature. The hotter the system, the more atoms are in the higher energy state. In fact, if you know what fraction of the atoms are in each energy state, you can plug that into the equation and solve for the temperature.
A laser operates by pumping energy into the atoms, pushing many of them into the high energy states. When the atoms drop back into the lower energy states, they give off the energy as a beam of laser light. But between the time they get pumped up and the time they drop back, they're in an abnormal state, with lots more atoms than normal in the high energy state. If you plug this abnormal distribution into the equation and solve for temperature, you may get a negative number.
Here are some websites that describe this idea in much greater detail: