An energy condition is one of several inequivalent but "similar" conditions demanding that the energy density is non-negative at each point of the spacetime. This is just a different way of saying that energy-wise, the vacuum is the emptiest environment you can have. See "energy condition" on Wikipedia.

There are several types of energy conditions, especially the null, weak, dominant, and strong energy condition, and some variations. Roughly speaking, they differ about the question whether the positive or negative pressure may be added to or subtracted from the energy density to make the inequality satisfied by weakening it or strengthening it.

Approximately but not quite rigorously, these four conditions are increasingly strong. A related fact is that they are decreasingly well established. In particular, the "null energy condition" is the weakest and nicest one and there are all reasons to believe that this condition really has to be obeyed everywhere in Nature. The others could perhaps be violated in some systems but no "truly realistic" examples are known.

The interpretation of the energy density and the inequality is obvious in a classical (non-quantum) theory. In the quantum theory, the energy density is an operator and one deals with various questions such as "whether one allows the energy density to drop below zero by random quantum fluctuations", "whether the inequality has to apply to all eigenvalues or just to some average in allowed states", and so on.

Some of the quantum refinements of the energy conditions may be violated by the Casimir effect, the apparently negative energy density (manifesting itself as an indisputable total energy and a force) due to the quantum character of the electromagnetic field in between two conducting plates. But if one calculates the average energy density along a curve of the flow, it's still true that the inequality expressing e.g. the null energy condition is obeyed for the averages.

These quantum subtleties only affect the accurate inequality verified with the precision of one Planck's constant or so.

But these quantum subtleties have no impact on questions about the motion of spaceships because spaceships are macroscopic objects and they require macroscopic energy densities. Consequently, the classical (non-quantum) version of the inequalities is enough to discuss whether the superluminal spaceship is possible and the answer is certainly No, they violate the null energy condition in a way that is not allowed, by terms much larger than Planck's constant.

One may be asked not to discuss the problems of warp drives with the causality explicitly in this question about the energy conditions. But one always discusses the causality issues implicitly because the energy conditions and causality are really equivalent, at least when it comes to the most typical consequences! Why is it so?

Consider a gas of particles (spaceships?) that are moving at a speed, possibly faster than light. You may derive the pressure of this gas. The pressure will be equal to p = –rho/3 for photons or, more generally, if the speed of the particles is the speed of light. If the particles only move in the x-direction, the xx-component of the stress tensor will be –rho. But if the particles will be faster than the speed of light, the absolute value of the pressure will exceed rho, the energy density, and one will violate the energy conditions!

So the energy conditions are "morally equivalent" to the speed limit on motion imposed at the speed of light. In quantum field theory, these two types of conditions are also equivalent to the stability of the vacuum. Tachyons, particles faster than light, would be quanta of a quantum field whose potential energy goes like –T^2 which may be negative (conflict with energy conditions) and is unbounded from below (implying instability, like an egg standing on its tip). In other words, tachyons could be pair-created in the vacuum, out of nothing, without violating any general laws. That would imply that the vacuum is unstable.

The warp drive papers like to assume that by a clever curvature of the spacetime, the speed limit (at the speed of light, c) imposed by special relativity may be circumvented in general relativity. Special relativity may be ignored, the authors suggest. But this isn't really a valid claim.

Special relativity applies in a general relativistic theory in various ways; general relativity is nothing else than a unique consistent theory based on special relativity that also includes gravity (force obeying the equivalence principle). One way how special relativity applies in a curved spacetime is "locally" because in a small enough region of the spacetime, the spacetime is nearly flat and Minkowskian, and special relativity therefore has to hold in this small region. This is the "resuscitated special relativity" that the warp drive theorists admit and want to circumvent in some way (colliding with problems with the energy conditions etc.).

But it isn't the only way how special relativity constrains the phenomena in a theory with the curved spacetime. The other way is at long distance scales, where the spacetime looks almost empty and flat again. When all the hypothetical spaceships are embedded in a nearly empty Minkowskian spacetime, there is still an unbroken Lorentz symmetry acting on this spacetime, and the spaceships only break the symmetry spontaneously and locally, as local perturbations. The Lorentz symmetry continues to apply to the theory of the "empty spacetime plus local perturbations" and it still implies that the faster-than-light motion is forbidden because it's equivalent to motion in the direction of the past, one that conflicts causality as specified in the special relativistic context.

Even if we have lots of black holes, objects with extreme red shifts etc., they may still be viewed as "particles" at distance scales much longer than their radii, and such particles have to respect the special relativistic speed limit, too. It's obviously critical for relativity that black holes (and anything else) can't get an exemption. After all, all elementary particles may be interpreted as "tiny black holes of a sort", even though black holes that demand large corrections to the low-energy Einstein's equations, and if black holes could get an exception from special relativity, then every particle and therefore everybody could get it, too!

There are no exemptions and particles, black holes, spaceships, and other objects simply can't move faster than light inside the surrounding Minkowskian spacetime. No amount of engineering attempting to compress or stretch the space, as long as it is allowed by the basic laws of Nature, can circumvent these conclusions because they are basic and universal conclusions of the 1905 special theory of relativity.