Energy conditions
General relativity on its own accepts any matter distribution. Physicists add “energy conditions” that ordinary matter is expected to obey. The weak energy condition says every observer measures a non-negative energy density. The null energy condition is a weaker version that is hardest to violate. Matter that breaks these conditions is called exotic matter.
Where negative energy appears in nature
In quantum field theory, the energy density at a point can dip below the vacuum value. The best-known example is the Casimir effect: two uncharged metal plates very close together attract each other because the energy between them is lower than outside. The effect has been measured. Quantum inequalities, however, limit how large and how long-lived such negative energy can be.
Why warp drives need it
Santiago, Schuster and Visser proved that physically reasonable warp drives violate the null energy condition, and therefore also the weak, strong and dominant energy conditions. Some observer will always see negative energy in the bubble wall, even if observers riding along see positive energy.