Since the weapon is built in a planet, where it's aiming depends on the planet's rotation.
Say I wanted to destroy a planet that was above[*] me (I'm evil like that!), how would I manage that?
[*] e.g. a planet from a star system that is directly above the base's North pole
We know it CAN be aimed, and very precisely. They were able to decide to destroy either entire starsystem OR one base on it (but leave other planets intact).
(in all fairness, the latter seems to contradict the point #4 below. Welcome to J.J. Abrams "canon" :(
Though visibly angry, Snoke’s tone remained unchanged. “Have we located the main Resistance base?”
Hux was clearly gratified to be the bearer of good news. “We were able to track their reconnaissance ship back to the Ileenium system. We are coordinating with our own reconnaissance craft in the area in order to lock down the specific location of their base.”
Snoke replied with cold satisfaction. “We do not need it. Prepare the weapon. Destroy their system.”
Collected and composed as he was, Hux was not immune to surprise. “The system? Supreme Leader, according to the most recent galographics, at least two and possibly three habitable worlds circle Ileenium. Following the destruction of the Hosnian worlds, would it not be worthwhile simply to destroy their base and claim the remainder for the Order? We will have the location of the base within a matter of hours and—”
Here's what happened when they fired the device. Note the bolded part which means that the planet itself was aiming the weapon by aligning its path.
Having said that, it says "artificial line of egress", without any further details - which MAY mean there is at least some flexibility in how that line is provided (assuming the line's containment is energy based, and not simply a long narrow hole drilled in planet's crust).
When the weapons engineers fired the device, a breach was induced in the containment field. At incredible velocity and accelerating exponentially, the concentrated volume of quintessence escaped, transforming as it did so into a state known as phantom energy and following the artificial line of egress that had been provided. Assuming that the rotation and inclination of the planet had been taken into account, the released blast of concentrated phantom energy would travel along a perfectly linear path, punching a small Big Rip through hyperspace itself until it left the galaxy —
— or encountered something in its path that was of sufficient mass to intercept it.
The weapon fires through hyperspace (or rather SUB-hyperspace).
Foster's novelization covers details of the weapon:
Wexley spoke up. “They’ve built a new kind of hyperspace weapon within the planet itself. Something that can fire across interstellar distances in the equivalent of real time.” His expression showed his incredulity. “I’ve had my share of technical training, but I can’t even imagine how that’s possible.”
This time Finn responded. “I can’t, either, but those of us assigned to the base heard rumors that it doesn’t operate in what we’d call normal hyperspace. It fires through a hole in the continuum that it makes itself. Everybody was calling it ‘sub’-hyperspace. That’s how it can arrive in moments across a distance like that between the base and the Hosnian system.
and
Finn nodded. “General Hux told us it’s the most powerful weapon ever built. He said that it can reach halfway across the galaxy.” Fresh murmurs of disbelief greeted this latest assertion. “And in real time. Because it doesn’t reach across the galaxy; it reaches through it.” He shook his head, which was starting to hurt from the effort of trying to explain what he had overheard but did not understand.
And here's how the actual shot at Hosnian Prime is described:
Traveling faster than anything ever generated by artificial means, through a torn portion of space-time whose properties were not fully understood, the concentrated glowing ball of energy lit the night sky above Republic City. ... It was as if a minuscule sun had suddenly appeared from nowhere, heading directly for the world on which she stood.
Slightly offtopic, but the speculation in another answer here that 4 separate beams were for 4 separate targets was NOT correct. What destroyed other planets in Hosnian system was not separate payload from the weapon, it was the explosion of Hosnian Prime:
...Turning the planet into what astrophysicists called a pocket nova.
Expanding outward from the explosion, a tremendous burst of heat tore through the Hosnian system’s other worlds, searing their surfaces clean of life and incidentally obliterating all settlements, installations, and outposts, as well as the hundreds of ships belonging to the Republic fleet. In its wake, the detonation left behind a blazing, spherical mass. The home of the Republic had become a new binary system: one utterly devoid of life.
Obviously you cannot change planet's orbit or positioning, at least without seriously affecting it's climate, gravity, and environmental conditions to extents where life on the planet is impossible or even you completely destroy it... so there must be some degree of control over the fired beam.
In fact, the scene where four planets are destroyed with a single firing of the weapon instills this idea, as what initially is a single beam splits into four different beams that seems to bend slightly their trajectories to reach different targets widely separated.
My guess is that there's some sort of "carrier" inside the beam that can hold the energy for the impact and guide or follow a determinate path to the target.
