While it might seem obvious what a planet is, the definition has become more complex since 2006 when the International Astronomical Union voted to reduce the status of Pluto (discovered in 1930 and considered a planet ever since then) to dwarf planet, so that it is officially no longer the ninth planet in the Solar System.
Under the new definition, a planet has to be large enough that its gravity forces it into the shape of a sphere (smaller, oddly-shaped asteroids therefore do not qualify); it has to be orbiting a star, and not a satellite of another planet (several moons in the Solar System are bigger than Pluto); it has to be not massive enough to cause thermonuclear fusion (in which case it would classify as a star); and it must have cleared its neighbouring region of planetesimals and other objects by its gravitational pull.
It was the last of these requirements that Pluto failed. Pluto is effectively part of the Kuiper Belt Objects, a wide belt of space towards the edge of the Solar System (between about 4.5 billion kilometres and 8 billion kilometres from the Sun) which contains millions of small, icy, rocky objects, of which more than 70,000 have been identified over 100 kilometres in diameter. Haumea and Makemake are two other dwarf planets in this region, and the dwarf planet Eris (which is actually larger than Pluto) is even further out, roughly three times Pluto’s distance from the Sun. Ceres, the largest object in the asteroid belt between Mars and Jupiter, is also technically a dwarf planet (which essentially means a small, more or less spherical planetoid, whose gravity is not sufficient to have cleared its local area of debris).
The distinction between dwarf planets, asteroids, meteoroids and comets is perhaps less clearly defined. Simplistically, asteroids are relatively small inactive bodies composed of rock or metals; dwarf planets are the largest asteroids; meteoroids are smaller particles of asteroids (called meteors or "shooting stars" when they burn up in the atmospere, and meteorites if they manage to penetrate to the Earth's surface); comets are mainly composed of dirt and ices rather than solid rock or metal, and tend to have dust and gas tails when close to the Sun.