Five terms, one table
The confusion usually comes from the three "meteor" words. They all refer to the same kind of object at different points in its journey.
Asteroid
A rocky or metallic body orbiting the Sun. Most are found in the asteroid belt between Mars and Jupiter. They range from dust-grain size up to 940 km (Ceres, now classified a dwarf planet). Unlike comets, they do not develop a visible tail. The IAU defines small Solar System bodies in Resolution B5 (2006).
Comet
An icy body — a mixture of ice, dust and rocky material. When a comet nears the Sun, ice sublimates and releases gas and dust, forming a coma (fuzzy atmosphere) and one or two tails pointing away from the Sun. Most comets originate in the Kuiper Belt or Oort Cloud. Halley's Comet returns every ~75–76 years. See our Dictionary entry for Comet.
Meteoroid → Meteor → Meteorite
A meteoroid is a small rocky or metallic body in space (typically <1 m; larger objects are classified as asteroids). When it enters Earth's atmosphere and burns up, the glowing trail is called a meteor (or "shooting star"). If any fragment survives to reach the ground, it is a meteorite. The American Meteor Society FAQ clarifies the naming in full.
Meteor showers explained
When Earth crosses the debris trail left by a comet, hundreds of meteoroids enter the atmosphere in a short period — a meteor shower. Because they all travel in parallel paths, they appear to radiate from a single point in the sky called the radiant. Showers are named after the constellation containing their radiant — the Perseids appear to come from Perseus, the Geminids from Gemini.
Check our Events page for the next major meteor shower, and the AMS annual calendar for exact peak forecasts.