'Negative pressure' counteracting gravity? Dark energy may be described as a fluid with negative pressure.
We say that this negative pressure counteracts gravity and accelerates the expansion of the Universe.
Now consider, for example, a star. Gravity contracts the star, but positive (thermal) pressure counteracts the collapse. 
This makes me confused because in both cases we have gravity as an inwards force, and in both cases we have pressure counteracting gravity, however, in one case it's 'positive' and in the other it's 'negative'. 
How can I reconcile this?
 A: Imagine you have a star sized ball of gas that is in equilibrium i.e. the pressure of the gas exactly balances the inwards gravitational force. Now imagine compressing the gas. This has two effects:
The first effect is the obvious one that compressing the gas increases the pressure, so the result is an outward force and if we stop compressing the gas we expect it to expand again. This is the standard Boyle's law behaviour that we all learned in school.
The second effect arises from general relativity. By compressing the gas we have done work on it, so the gas/star now has more energy than it did before. But in GR energy causes curvature just like mass does. In fact the stress energy tensor doesn't distinguish between mass and energy - it uses a single value for mass/energy density using the famous formula $E = mc^2$ to equate the two. So by compressing the gas we have increased the spacetime curvature caused by the star so we have made it's gravitational field stronger.
Under most circumstances the increase in the gravity caused by pressure is insignificant, and pressure has the effect we expect i.e. it causes the star to expand. However in extreme cases, like the neutron star collapse mentioned by Chris in his comment, the gravitation effect of pressure overcomes its effect on expansion and pressure then contributes to the collapse.
Dark energy is rather different. I started by pointing out that by compressing the gas and increasing the pressure we are doing work on the gas/star and this contributes to its gravity. If you take some region of vacuum containing just dark energy and compress it the dark energy does work on you i.e. the energy of the space you've compressed decreases. This is what we mean by a negative pressure. Because of this dark energy can't cause attraction in the way that pressure can.
A: You should distinguish two things: the pressure itself, and the gravitation that it creates. First of all, why does it create some gravitation? Because energy does. It is not the mass of the Earth that pulls us down - it is the energy of that mass, $E=mc^2$, that makes us heavy and not flying but walking.
And the energy does that indirectly. First, it creates, "emits", a gravitational field, and next, that gravitational field tells us where to be pulled to. That is different from anything that energy could do to you directly - for example, acting directly, energy could heat you.
And the same difference holds for the pressure itself (its direct action) and its gravitational effect. For example, you can pull a spring. Then your hands would feel its tension directly. And if you stand near someone who pulls the spring, you would feel its indirect gravitational effect (it would be smaller by factor $c^{-1}$ of several orders, and it would attract or repel you from the spring).
That is what happens with the Dark Energy. It has negative pressure, and could pull us with it, but we don't feel it. It is compensated - the same way as we don't feel the atmospheric pressure, and deep sea fish does not feel the pressure too. But its indirect gravitational effect still remains. And we observe it by the overall pull of galaxies from each other.
The sign of gravitational effect is found by the sign of the usual gravitational effect of our positive masses and energies. The theory should be symmetrical with respect to time and space, and similar temporal and spatial quantities. Because of that, positive energy attracts; positive pressure attracts; and negative pressure, or pull, repels.
A: Understand you could keep the "Dark Energy" (= Cosmological Constant) term where Einstein originally inserted it, on the LHS of the equation, and view it as an intrinsic part of the Curvature Tensor.  If so, the "empty" space-time fabric would not be perfectly "flat" (Minkowskian) but would have an intrinsic curvature, vaguely like the intrinsic upward bowing of a flatbed trailer, which bends upward when empty so as to support heavier loads.
Would like to speculate whether perhaps the "missing polarity" of gravity (negative mass) somehow resides within the "fabric" of spacetime itself?  Instead of matter possessing both polarities of charge (plus & minus, obeying "opposites attract" and "like repels like"), perhaps all matter has one polarity of mass (plus) and all spacetime fabric has the other (negative)?  Perhaps, too, "like attracts like" and "opposites repel" with Gravity?
Could a spacetime fabric possessing a "material property" of "negative mass" which repels normal mass be consistent with Dark Energy?
