Heat as a slang term is commonly used to describe a weapon of some kind, usually a pistol. "You packing heat?"
Heat as in heat generated by an exothermic reaction is an increase in energy of particles causing them to move/vibrate more rapidly, in some materials this can cause electricity to flow more easily, for materials to melt then evaporate.
When an animal "goes into heat" they are at their most sexually active and reproductive time of the year, this would usually occur during the "mating season" of the particular animal.
As a military abrieviation, HEAT means High Explosive Anti Tank. This is a rocket or other explosive missile/shell designed to penetrate through a tracked vehicles armour before detonating it's explosive charge to disable/destroy the target. (For those of you wondering, there is such a thing as "low explosive")