Our solar system has one star, eight planets, several dwarf planets, a whole bunch of moons, lots of asteroids and comets, and a great many small chunks of ice, rock, and dust. Astronomers have known about the Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn since ancient times. Other parts of the solar system were gradually discovered after the telescope was invented in 1608, including Uranus, Neptune, and Pluto.
More recently, astronomers continue to discover and calculate the orbits of smaller pieces among the asteroids, in orbit near Neptune's path, and beyond, then give them names from various ancient religions and mythologies. Pluto turns out to be one of a great many objects in the Kuiper Belt - like the asteroid belt, but beyond Neptune and mostly made of ice rather than rock. This led to a 2006 reclassification of all solar system objects that orbit the Sun, based on (roughly) size - or more precisely, certain mass and gravity thresholds required to do things like be round and clear other objects out of its orbital path.
Our solar system, then, consists of everything within the bounds of gravitational influence from the Sun. Aside from the more well-known parts nearer to Earth, there's the Kuiper belt and "scattered disc" which contains all the trans-Neptunian objects (with orbits that cross paths with Neptune), then the Oort cloud that goes all the way to the edge, where astronomers believe comets come from.