Planets (from Ancient Greek ἀστὴρ πλανήτης (astēr planētēs), meaning "wandering star”) are astronomical objects orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, but is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.
The term planet is ancient, with ties to history, science, mythology, and religion. The planets were originally seen by many early cultures as divine, or as emissaries of deities.
As scientific knowledge advanced, human perception of the planets changed, incorporating a number of disparate objects. In 2006, the International Astronomical Union (IAU) officially adopted a resolution defining planets within the Solar System.
This definition is controversial because it excludes many objects of planetary mass based on where or what they orbit. Although eight of the planetary bodies discovered before 1950 remain "planets" under the current definition, some celestial bodies, such as Ceres, Pallas, Juno, Vesta (each an object in the Solar asteroid belt), and Pluto (the first-discovered trans-Neptunian object), that were once considered planets by the scientific community are no longer viewed as such. These smaller planet-like objects of which there may be hundreds are called dwarf planets.
By Richard Conan-Davies
The question of what a planet is, is something that astronomy has struggled with for quite some time. A planet is essentially any large naturally round object in orbit around a central star, that has cleared it's orbital path of other smaller sized objects called planetesimals. It also can't have a thermonuclear reaction that is self sustaining like a star has.
A planet is round naturally due to it's large size. Gravity forces all the parts of the planet into a spherical shape (sometimes this is called hydrostatic equilibrium).
But this is not quite enough. If there are still other small objects ( like the asteroid belt or Kuiper belt) that drift around it's orbit of a similar size or mass, it is referred to as dwarf planet. Pluto is a good example of a dwarf planet.
Note: Here is a more technical discussion about ‘clearing the neighbourhood’ from wikipedia
Our solar system officially contains 8 planets. Officially means, according to the International Astronomical Union. Practically and for exploration purposes though there are hundreds of planet like things orbiting the sun in our solar system.
In addition to planets in our solar system there are also planets around other stars. These are called exoplanets. Exoplanets could be even more complicated classifying because some systems may be very young and have active accretion discs. These discs have lots of varying sized objects collide and produce proto-planets that may have very weird stretched out structures.
A general term: Planetary mass objects
A more general term that is sometimes used is a planetary-mass object (PMO), planemo, or planetary body. This covers any object that is essentially large, round not sustaining fusion but could be anywhere and may even include moons or rogue drifting planets.