Simple answer, it's technically a concentrated beam of pure energy, aka an ultra powerful laser.
But how can that destroy a planet?
In the video clip we see that there are five beams that combine to make a single solid beam. Using constructive interference the amplitude of the energy beam could, in theory, be enhanced - much like a AN/SPY-1 Phased Array can cook a seagull from a mile away without touching the bird flying next to it!
So let's say the laser beam consists of some sort of phased array that concentrates energy along the beam's path at certain points. How much energy would be required to blow up a planet. Alderaan is roughly the size of Earth.
It would take somewhere around 2.24 × 1032 Joules of energy to cause the Earth to explode. That's a lot. So our beam needs to be able to concentrate that much energy into a point.
We also need a type of energy that penetrates through the planet and deposits the energy into the center of the planet - if it was a normal laser beam it would deposit that energy on the surface which would vaporize a hole through the atmosphere and ground, but would take MUCH longer to blow up. We need the energy to explode the planet (like in the clip).
We'd also want our energy to heat the metallic core, but not the surrounding rocks that much. This will result in the best style explosion. Heating the rocks would be a waste of energy because we just want the planet to 'pop,' not heat up and melt.
So our laser beam needs to focus roughly 2.25x1032 J into the metallic core. We can assume at this point they are using electromagnetic waves outside the visible spectrum (so it wouldn't really be green).
Neutrinos are about the only thing that could be produced en masse and could pass through the Earth. The problem, they don't react with anything (which is why they pass through the Earth). Gamma rays interact weakly with the atmosphere, but on the scale we're talking about, they'd ionized and destroy the atmosphere as well as all the rocks and soil - they wouldn't just penetrate the ground.
What's left? Well, some sort of dark energy is probably what must be used. Dark energy can interact with regular matter by imparting, but doesn't seem to be like normal energy in that it penetrates and is barely detectable. So let's say it's a dark energy beam.
Okay, so five smaller dark energy beams, phased to constructively focus energy at a predetermined point, combine into one giant dark energy beam. That beam must be on the order of 2.25x1032 J of energy.
To put that into perspective the sun produces about 3.86 x 1026 Joules per second (or about 10,000x more energy than our Death Star). Maybe the Death Star roams around with a little star (~10,000x smaller than the sun) inside it.
The Death Star could then be a Dyson Sphere designed to capture all the energy from a small star. That would actually explain it's shape. However, the Death Star is too small to enshroud a normal small star. The star would actually need to be ultra dense (the Death Star is only about 120km across). Therefore, the Death Star would actually be surrounding a Neutron Star.
Ah! Now we're getting somewhere. A neutron star is a collapsed star that is made entirely of neutrons (the protons and electrons were squeezed together under crazy intense gravity!).
Neutron stars also spin incredibly fast. With a strong magnetic field it's not wholly unbelievable they create tremendous power. Let's say this is what they do.
So we've got a Death Star with a neutron star heart powering a dark energy phased array planet killing beam. This is probably how it works in real life.