How do the terms comet, asteroid, meteoroid, meteor and meteorite 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.

http://www.nearearthobjects.co.uk/

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.