Low temperatures change the speed of different chemical reactions, throwing delicately balanced systems out of balance. How resilient a creature is to temperature changes is largely a function of how complex it is, and how buffered it is against this kind of change.
Tardigrades as kmm mentioned essentially can't be killed by cold, surviving down to almost absolute zero (absolute zero ≈ minus 273.15°C). Most unicellular organisms can be frozen (especially if frozen quickly or cryoprotected with something like DMSO) and revived later without any change. Most strain libraries of bacteria are kept frozen for cost reasons. Even human cell cultures can be frozen and thawed, even though some of the cells will die. (If even 10% of your cells died you as a person would probably die, so freezing is off the table unless you are very clever indeed.)
Humans and most mammals are pretty sensitive to temperature changes, even though reptiles are not. Most fish are cold-blooded and are adapted to cope with the relatively cold temperatures, but the human heart is not well adapted to even relatively warm temperatures. The atria and the ventricles of the heart stop communicating below 27°C (80.6°F)or so, and below 20°C (68°F) the heart can stop beating entirely. Nerves are complex systems, and absent regulation systems in the nerves themselves to compensate for the changes in temperature, they can stop firing completely. A simple heart can run with less sophisticated orchestration, and is, therefore, more resistant to temperature change. A complicated heart involving the coordination of the atria and the ventricles to produce effective pumping action is affected much more strongly by the drop in temperature.
So, in general, more complicated implies more vulnerable to temperature changes. There are exceptions, however. Bears can hibernate and drop their core body temperatures without any ill-effects, so it's possible to adapt to. Fish deal very well with cold temperatures if exposed gradually, and some fish could live quite happily in a slushie, salinity notwithstanding.