If something could realistically happen, it's a possibility. So winning the lottery is a slight possibility, but becoming king of the world is not even a remote possibility.
In one sense, a possibility is an option: "One possibility is that the UFO is just a shooting star." It can also mean more of a potential: "This house might look like an abandoned shack, but it's got so much possibility!" All meanings of possibility share the sense that something is able to happen, or is possible. The Latin root is possibilis, "that can be done."