First, how much energy does blowing a planet up require?
What holds a planet together is gravity, and the gravitational binding energy of Earth is about 2.5 * 10^32 J. So an approximation, that's how much energy you need to turn the Earth into a bunch of debris. Of course, lighter planets like Mars require less energy, and a super-Earth requires more, at those scales, it won't make that much of a difference anyway.
There will also be losses, so let's round this up to 10^33 J. That's about the total energy output of the Sun in a month. Putting so much energy in a 1-second laser is going to be a problem, though not a physically impossible one.
When you want the biggest, baddest laser possible, the obvious solution is a Nicoll-Dyson beam - that is, a Dyson sphere acting as a giant laser emitter. Using short enough wavelengths, it can very well strike planetary-sized (or even smaller) targets. Problem is, it would need to store energy for a long time - for example, making a few trillion tons of antimatter - and then release it all in one second to generate the laser. Or they could a giant interstellar energy network in order to power a laser emitter. One option is to use the central galactic black hole as an power source (by throwing stuff into it) and interstellar gas clouds as maser transmitter.
Note that the emitter or transmitter will most likely be destroyed by the heat losses. Even if those are very small, a small percentage of apocalyptic energies is still a lot.
Anyway, we have our laser emitter, and it fires a 1-second long beam of 10^33 J at the planet. Then, weird things happen. The details would require simulations, but we can figure out the main effects.
When so much energy hit the atmosphere, it will instantly turn into plasma. If air is transparent to this wavelength (like, say, visible light), enough will hit the ground in the first millisecond to hit it with the force of countless nuclear weapons, but in any case the entire local atmosphere will turn into plasma and thus most of the second-long beam will be absorbed by what was formerly known as the upper atmosphere.
Air will also undergo nuclear fusion, which would add a bit more energy, which may or may not be negligible compared to the laser energy. There will even be some matter-antimatter pair production. This means that whatever the initial wavelength, some X- and gamma rays will be sprayed around, including back to space.
Plasma is opaque, and the laser would more or less be entirely absorbed by it before hitting the surface. That's similar to a nuclear fireball where most of the X-rays are more or less instantly stopped by the atmosphere, which then carry the energy as a giant fireball.
The plasma fireball will then do three things: it will reflect some of the laser back to space, it will expand as a shock-wave and it will radiate energy back all around. The planet is concerned by the last two.
The light will be strong enough to turn even more atmosphere into plasma, as well as a big chunk of the ground. Further away, it will be strong enough to instantly vaporise anything it lights. So expect the continent that is hit to instantly be caught into the plasma ball, and the hemisphere to be flash-vaporised by the fast-expanding plasma ball.
Due to the slight lens effect and the scattering of the atmosphere, the other hemisphere would also be exposed to the light. I would expect everything at the surface to also be instantly be burned, though the antipodean point my be slightly better off. Underground bunkers may also survive, if the detonation of the entire surface flash-vaporising doesn't compress them too much.
So yeah, a million voices suddenly cry out in terror and are suddenly silenced. And many more don't even have time to cry out.
Then, the fireball expands, and causes a giant shock-wave. At those scales, a planet acts like a big blob of liquid (hence its round shape), so it will look like a giant tsunami, but where the crust and upper mantle is making the wave instead of simply the ocean. The shock-wave is also travelling through the interior of the planet, so it will reach the other side before the surface wave. Doing so will cause earthquakes and volcanoes, in a sense that the Halifax explosion made a noise. That is, the crust will break apart and the mantle splash around. I would expect that to take around a few minutes, ten or twenty minutes at most.
This is also the point where the planet starts breaking apart, a bit like a big water drop being hit very hard. It fragments in gobs of varied sizes and shapes, from dust grains to, possibly, some as big as the Moon or Mars. Once settled, the big ones will settled back in round shapes.
Almost all of them are extremely hot, and will stay hot for a long time. Asteroid-sized ones will take years or centuries, the biggest ones may take millions of years to stop glowing - you can look at the history of our planets for details there.
In addition to the atmosphere and hydrosphere being blown to space, a chunk of the planet has been vaporised, and is also blown to space at great speed. In addition, the mantle and core were under tremendous pressure, which is suddenly released. This will make them splash back and/or vaporise with a vengeance. All this will cause the rest of the planet to be pushed in the other direction, as if the planet was a big laser-thermal rocket.
The fragments will, for the most part, still end up in a solar orbit. The push would have been enough to make the orbit vary a bit if they all stayed the same (the details of the new orbit depending on which side of the planet was hit). However, the fragments will not all stay in a roughly Earth-like orbit. Some will have been violently ejected by the energy of the laser, or by how the shock-waves interacted. Other will be destabilised by gravitational interaction with the other fragments.
Starting a few centuries or millennia down the line, and possibly lasting millions of years, the other planets of the system can expect nasty meteoric impacts, some very fast and some very big. Not enough to break another planet apart, but enough to definitely ruin most planetary surfaces at some point.
But even before that, those planets have other problems.
Anything not in the shadow of the planet will have been exposed to the light of the laser being scattered at impact. This scatter, for one second, is emitting as much energy as about a million times the Sun.
If there is something like the Moon and it was in the shadow of the planet, it will take a fair amount of debris in the face, probably breaking bits of it and sending them flying as well. The entire surface is, of course, turned upside-down into a lava ocean.
If this Moon is exposed to the flash, the exposed hemisphere literally explodes into plasma. The vaporised plasma pushes it in the other direction and send it in a slightly more different orbit than the debris. The shock-waves will turn the other hemisphere upside-down as well, and debris will bombard it for a long time, probably starting the following years if not sooner. Again, bits may be sent flying.
The exposed hemispheres of more distant planets will be burned to a crisp, as if by a close nuclear explosion. Atmospheres will be partially blown away. Ice surfaces will explode into steam, some of which may keep around as atmosphere. Surfaces may be vitrified. On the non-exposed hemispheres, all this may also cause earthquakes. The details depend heavily on the planet type and its distance from impact.
There won't be much effect on the local star(s), though weather patterns may be affected in weird ways. Maybe there will be more solar flares in the future, though at that point, few would probably care.
Exposed neighbour star systems will see the flash easily. At a million times the intensity of the Sun, assume a visible wavelength for the laser, anyone looking at the sky will have a hard time missing it, even for a one-second flash. It will be as bright as a thin Moon crescent, but concentrated in a single point, and with a distinct colour.
With access to modern astronomical instruments, probably half the galaxy can see it (by the time the flash arrives, obviously), and possibly neighbouring galaxies as well if they're lucky. Between a very distinct signature and the probably hard-to-miss stellar engineering next to it, it won't take long for them to guess what it was.
Special mention for the Death Star:
Star Wars is apparently taking place in an universe with different physical laws than ours. There is pressure in the vacuum (hence walking on an asteroid with only oxygen masks, flying around as if in atmosphere, space fireballs when things explode...) Flying requires very little energy (no visible radiators or incinerating exhausts). Planets may be smaller, and stars are. Asteroid fields are much, much denser. Star systems are closer together.
In this case, planets are probably much easier to blow up, and their debris cool down near-instantly afterwards.
