Recently, a group of physics students at the University of Leicester wrote a paper detailing the physics of the Death Star, and estimating the amount of power it would have to output if it was able, as shown in Star Wars, to be able to totally destroy a planet, turning it into nothing but rubble. The steps to finding the answer are not that complicated, since it is just finding the gravitational binding energy of the earth (or a similarly-sized terrestrial planet) and then concluding that the Death Star must be able to fire at least that much energy.
Which, of course, leads to other questions. There are several questions about the physics of the Death Star that need to be answered. According to figures I have read on the Star Wars wiki, the two Death Stars we see in the films were 160 km and 900 kms in diameter, therefore it is likely that they would produce enough internal gravity that the normal structural strength of steel or other alloys would fail. However, since the Star Wars universe possesses artificial gravity, it seems that they would also possess artificial anti-gravity. So this is a rather minor problem to hand wave away.
Two issues more directly related to the power output problem are fuel consumption and gravitational tidal forces. According to the paper, the energy output of the Death Star is on the order 10^32 Joules. The most efficient (known) energy production source is the annihilation of matter and anti-matter, and to produce that amount of energy, about 10^12 tons of anti-matter (and matter) would have to be annihilated every time the Death Star fires its main weapon. This would amount to carrying around several cubic kilometers of anti-matter per shot. However, the Death Star instead uses hypermatter reactors which are a form of unobtanium. Another issue that I briefly considered is that the beam of energy that the Death Star sends out would be energetically dense enough to create tidal forces on nearby bodies. Energy has a gravitational field, and an energy beam that was narrow and energetic enough would create tidal forces, as it pulled more strongly at nearby objects than at ones further away. I was thinking of doing the math for this, but I realized that there is a much more obvious physical problem.
Each time the Death Star fires, it creates a beam strong enough to destroy the earth, an amount of energy that is, at a conservative estimate, something like a million times stronger than the entire energy reserves of the earth's fossil fuels. As much energy as the sun radiates in a month. All of it being channeled through the electrical system and conduits of a space ship, that while large, is still a million times smaller than a star. Since the beam only fires for a few seconds, it is dealing with all that energy in a much smaller space and time. There is no heat sink or cooling method that could possibly prevent some of that energy from leaking out and turning the entire base into slag. There is no electrical wiring that could carry that much energy, and even a system of mirrors and light would shatter. In other words, the biggest impossibility in the Death Star, from a physics standpoint, is from the Second Law of Thermodynamics.
Of course, one of the assumptions about this is that the Death Star's weapon simply outputs enough energy to destroy a planet. It could also be that it only acts as the trigger for a chain reaction. Perhaps there is some technology that speeds up the natural radioactive decay inside a planet, or maybe the weapon temporarily suspends gravity, causing the planet to blow itself apart.
An even more important possibility to be examined is that Star Wars is not a science-fiction movie, but rather a fantasy in space, and if you watch Return of the Jedi and spend more time thinking about the physics of the Death Star than Princess Leia in a metal bikini, you are totally missing the point.