A meteoroid is a small rocky object in space in the solar system, smaller than an asteroid. A threshold is sometimes used to distinguish the two, but the traditional difference was a meteoroid is whatever produced the streak of light seen as it passes through Earth atmosphere (the streak of light is termed a meteor, and the term is also perhaps used for the object itself during that brief period), and an asteroid is an astronomical body observed by telescope. I've seen thresholds listed as 1m, 4m, 10m, and 100m, undoubtedly influenced by whatever issue is being considered. The lower end of meteoroid sizes extends to microscopic (i.e., a micrometeoroid).
A meteorite is the remnant of a meteoroid that has impacted Earth. The similarity of terms has led to mixing in usage (i.e., inaccurate, according to the above distinctions): meteor is sometimes used for any of the three, and perhaps meteorite for the object still in space. And when, for an example, a micrometeoroid imbeds itself in an artificial body while still in space, I don't know what the proper term is.
A meteor shower is a bunch of meteors occurring over a short time: meteor showers are predictable as they occur when the Earth passes through a region of the solar system where a belt of meteoroids orbit, generally considered debris from a broken comet or asteroid. Such showers return the same time each year, appearing to come from the same point in the celestial sphere, a point known as the shower's radiant or apparent radiant.