These terms are frequently used interchangeably by the uninitiated to mean approximately a “space rock”. In practical terms, how do their meanings differ?
At the risk of being snarky (each definition is from wikipedia)...
Comet - A comet is an icy small Solar System body that, when close enough to the Sun, displays a visible coma (a thin, fuzzy, temporary atmosphere) and sometimes also a tail. These phenomena are both due to the effects of solar radiation and the solar wind upon the nucleus of the comet. Comet nuclei range from a few hundred meters to tens of kilometers across and are composed of loose collections of ice, dust, and small rocky particles.
Asteroid - Asteroids (from Greek ἀστήρ 'star' and εἶδος 'like, in form') are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones. These terms have historically been applied to any astronomical object orbiting the Sun that did not show the disk of a planet and was not observed to have the characteristics of an active comet, but as small objects in the outer Solar System were discovered, their volatile-based surfaces were found to more closely resemble comets, and so were often distinguished from traditional asteroids
Meteor - A meteoroid is a sand- to boulder-sized particle of debris in the Solar System. The visible path of a meteoroid that enters Earth's (or another body's) atmosphere is called a meteor, or colloquially a shooting star or falling star. If a meteoroid reaches the ground and survives impact, then it is called a meteorite. Many meteors appearing seconds or minutes apart are called a meteor shower. The root word meteor comes from the Greek meteo¯ros, meaning "high in the air".
I am sure the above answers will serve well for the OP, but I tend to remember it simply by this :
Asteroid: A relatively small, inactive body, composed of rock, carbon or metal, which is orbiting the Sun.
Comet: A relatively small, sometimes active object, which is composed of dirt and ices. Comets are characterised by dust and gas tails when in proximity to the Sun. Far from the Sun it is difficult to distinguish an asteroid from a comet.
Meteoroid: A small particle from an asteroid or comet orbiting the Sun.
Meteor: A meteoroid that is observed as it burns up in the Earth's atmosphere - a shooting star.
Meteorite: A meteoroid that survives its passage through the Earth's atmosphere and impacts the Earth's surface.
First the simple distinction: both comets and asteroids are in orbit around the Sun, and sufficiently far away from the Earth that they move quite slowly through the sky. You need to look really closely (with a telescope) to see them move against the starry background. Meteors start out also in orbit around the Sun, but they are very small so we can't see them until they enter the Earth's atmosphere. They are travelling very fast relative to the Earth, which makes them heat up, so that we view them as rapidly moving streaks of light across the sky. A meteor takes a second or two at most to cross the sky, while a comet or asteroid takes weeks or months. There are also artificial satellites in orbit around the Earth. These move at an intermediate speed, taking about 5 or 6 minutes to cross the sky.
So, it's very easy to tell the difference based on the speeds they travel: meteors fast, satellites intermediate, comets and asteroids slow.
Now the hard part: comets vs. asteroids. Comets are made up mostly of ice. As they near the Sun, they are warmed and the ice sublimates, forming a tail which is swept away by the solar wind. Asteroids have already visited the region of the Sun many times in the past, and have lost most of their ice, so they don't form tails. In a telescope, they look just like stars, except that they move from one day to the next, like a planet.
The three classes of objects grade into each other with no clear dividing line. Classically, a comet gives off a lot of gas and dust, even forming a noticeable tail. Once the volatiles are mostly used up, an inactive comet can be "mistaken" for an asteroid, except perhaps on a close pass to the Sun when it heats up enough to outgas again. There are many asteroids out there (particularly those that swing through the inner Solar System) that probably started life with a lot of ice and would have been called comets, but are now almost entirely rock and dust.
There is no clear line between an asteroid and a meteoroid. Both are rocky/dusty; it's a matter of size as to how you refer to it. However, there is no clear-cut dividing line. A big one is usually referred to as an asteroid, while smaller stuff is a meteoroid. You could draw the line on whether it's big enough to likely survive a trip through the Earth's atmosphere (as a meteor) and arrive on the surface as a meterorite, but that also depends on the composition, initial speed, and trajectory, not just diameter. And of course, it would depend on which planet (gravity and atmospheric density) it's heading towards.
An asteroid is a small, natural object in orbit around the Sun, which has no history of ever displaying cometary appearance. "Planets" are large enough to never be considered asteroids. The International Astronomical Union also defines "dwarf planets" which are smaller than planets, but bigger than "asteroids." The qualification "natural" must be added because the number of man-made objects in orbit around the Sun are also never considered asteroids.
If an asteroid is ever observed to have a coma—a surrounding cloud, or a tail—a coma extended to have a linear appearance, it is officially categorized as a comet, and ceases to be officially categorized as an asteroid. Scientifically, we know that the usual reason for this appearance is an icy composition, but it is much more difficult to determine composition than to observe appearance so most objects categorized as comets remain categorized solely on appearance.
A meteor is a small (natural) object which is (or could be) observed entering the atmosphere. The countless meteors visible any given night are almost always smaller than any cataloged asteroid.