Exotic energy is not really definable and is more or less generic but however I will elaborate, exotic matter\energy is any object or entity which is not either present at our current knowledge or is an hypothetical and not proven scientifically or is rare to discover or maybe unfortunately can never exist and will remain our dreams. To be honest I could even consider anti-matter to be exotic as its not commonly encountered.
Its too generic to be given an set-solid definition for.
Negative energy\mass is again hypothetical as there are no proof of existence of such energy or mass thus is considered exotic. Nevertheless, let me elaborate on its effects and its hypothetical behavior.
Einstein describes all mass with positive value will "press" down upon fabric of space-time however "negative" shows all of the Negative mass to have negative mass (as straightforward as it sounds...) this means it should do an opposite as creating gravity - Negative gravity so unlike pulling it will push mass away from its space so that's one behavior of negative mass.
So in other words its like anti-gravity and due to this property its an ideal material for development of hyper-crafts and other sci-fi based technologies. However since its not or has been observed in space nor on an laboratory it still remains in the sci-fi category.
Final words, mathematics permits all kinds of unreal actions taking place such as being able to travel faster than $c$ (speed of light) however in physics they need to proven and tested therefore many physics concepts may be eligible and logically sensible in the beautiful language of mathematics but reality may not permit these things, therefore these may objects proven in mathematics and may never co-exist in our universe.