Some celestial objects seem to be referred to as asteroids, some as comets, and some as meteors or meteorites.
What is the distinction between all of these different objects? Are any of them the same?
The objects you are refering to are actually two different objects: asteroids and comets. Meteor and meteorite are other names for an asteroid, at a given time of its interaction with our planet. We'll get to that.
So first, what is the difference between an asteroid and a comet?
A comet is a small solar system body that display a "coma" (an atmosphere of a sort) and sometimes a tail passing close to the Sun. They are mostly made of ice and dust, as well as some small rocky particles. We distinguish two kind of comets, with short or long orbital period. The short orbital period ones originated from the Kuiper Belt, a region composed of small bodies beyond the orbit of Neptune. The long orbital period ones originated from the Oort cloud, a scattered disk of icy planetesimals and small bodies laying around our solar system.
An asteroid is a small body, composed mostly of rocks and metals. In our solar system, they can be originated from the asteroid belt, laying between Mars and Jupiter, or from the orbit of Jupiter (the Jupiter Trojans), or from actually almost everywhere in the Solar System.
An asteroid that enters the Earth's atmosphere becomes a meteor, what we also call a "shooting star".
Eventually, a meteor that was massive enough not to be completely distroyed entering the Earth's atmosphere and hitting the ground is a meteorite.
Asteroids are what are called minor planets - small inner solar system rocky bodies. Especially between Mars and Jupiter.
Comets typically have highly elliptical orbits but they don't have to. The key defining feature of a comet is that it has a gaseous atmosphere and a tail when it passes close to the Sun. Comets are generally loose aggregations of ice, dust, rocks etc.
Meteoroids are very small rocky or metal bodies, from a single particle up to about 1m diameter. If they hit the atmosphere, the visible glowing trail of particles is called a meteor. And if it hits the ground it is called a meteorite.