An object smaller than 5 m is called a meteoroid because it would "look like a meteor" if it fell in Earth's atmosphere. Strangely enough, the pieces that do survive to the surface are called meteorites, even though they must come from asteroids, not meteoroids (by definition, a meteoroid should not survive all the way to the surface).
Before people accepted that meteorites came from space, they were often called aeroliths (lith = stone, aero = air -- aerolith = rock from the air).
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Comets are very different.
They were already known to the Greeks who called them aster cometes (stars with hair). They are chunks of ice (actually a mix of stones, pebbles, dust and different types of ices). As they get closer to the Sun, they warm up and the ice vaporizes, giving the comet a "head" of gas and dust.
Pressure from sunlight pushes some of the gas and dust away from the head (this is why the tail always points away from the Sun). UV rays will also ionize some molecules and these will react by shining light of their own (same idea ss the Norther Lights) and this is called the ion tail.
Because the ion tail is made of charged particles, and because charged particles can be affected by magnetic fields, the dust tail and the ion tail are often separated by a few degrees.
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meteor = streak of light in the upper atmosphere
asteroid = chunk of rock, smaller than a planet
comet = chunk of ice(s) that reacts to the heat of the Sun.