A fireball is a bright meteor that could result in a meteorite falling to the ground.
A meteor is a streak in the sky caused by a meteoroid entering our atmosphere, usually at more than 70,000 km/h. Meteors are sometimes colloquially called a shooting stars.
A meteorite is a rock that has fallen from space and survived the passage through the atmosphere to land on Earth. Not all meteors survive to exist as meteorites. Most burn up completely in the atmosphere.
A meteoroid is an ancient piece of our solar system, too small to be classified as an asteroid. The distinction between a meteoroid and an asteroid is debated and fuzzy, but one rule of thumb is that if we can observe the object with ground-based telescopes, it is an asteroid. Objects too small to be detected by telescopes are meteoroids.
A meteor shower is a significant number of meteors happening about the same time when our planet passes through a stream of interplanetary dust and debris left behind by a comet or some other source. This results in a number of meteors appearing to come from the same general part of the sky, or radiant.
A comet is not a meteor nor a shooting star. A comet is a solar system body in an eccentric elliptical orbit around the Sun or that passes the Sun in a hyperbolic orbit. A distinction is made between asteroids and comets in that comets have an extended atmosphere around their nucleus and are believed to have a different genesis. Comets tend to leave debris in their path and the interaction of this debris with our atmosphere can result in a meteor shower.