Starkiller Base is the next generation Death Stars that have been developed by the Galactic Empire, now known as the First Order. Whereas in the past, these giant space stations have been fully artificial, this new one was once a planet in much the same way Hoth and Tatooine was. What the First Order have done this time round is to take a planet and hollow it out and turn the actual planet into a weapon.
General Hux is the commander in charge of the base, in much the same way Grand Moff Tarkin was in charge of the first Death Star which was destroyed by Luke Skywalker. Talking of which, the name Starkiller was to be the original surname for Luke Skywalker before Lucas had a change of heart and decided to go with Skywalker instead.
The Death Star in Star Wars IV - A New Hope was famous for blowing up Alderaan, a planet, this new one however is powerful enough to blow up whole Solar Systems. The First Order demonstrate the power of this new machine by destroying Hosnian Prime, a planet that is the new seat of government for the Senate which stands for everything that the First Order doesn't.
As its a planet, it has a breatheable atmosphere like Hoth which was the first Ice World to be featured in the Star Wars films (Star Wars V - The Empire Strikes Back). The planet will play host to being the location for the climax between the Rebellion and the First Order. Han Solo's character who is rumoured to being killed off in Force Awakens but does appear on the planet by the looks of revealed photographs.
The name also derives by the fact that the planet draws its power from nearby stars there by affectively killing off the star. Once the weapon has received the necessary power, it then fires its beam across the galaxy. Like the Death Stars of the films of old, this planet doesn't last more than one film.
It is the location of the death of one of the main characters in the series of films, that is Han Solo who is killed by Kylo Ren when Han confronts his son to try and turn him back to the side.
Supreme Leader Snoke is not on the planet at any time and is therefore able to appear in future films. It is unclear as to whether Kylo Ren and General Hux who were both on the planet at the time as to whether they managed to get off the planet. However given the habit Star Wars has in that they kill off lead antagonists in the first films of the trilogies, it is good bet that the lead First Order officers are dead.