With very few exceptions, meteorites come from asteroids; sometimes we even know which ones. They formed from the same material that was present at the birth of the solar system - but didn't "clump" together to form the planets.

The most common form of classifying meteorites is by how and where specifically they were formed. The numbers in the classes indicate how much melting/vulcanism the material has undergone - from 3 (none, with large chondrules) to 6 (melted, with chondrules absent).

Chondrites were the building blocks of nearly everything in the early solar system. In a zero gravity environment, minerals in the solar nebula coalesced into (mostly) spheres called chondrules. Chondrites are made up of chondrules mixed into other solar nebula material - like blueberries in a muffin. As asteroids formed, chondrites are the material that remained on the surface of an asteroid and have not melted since their formation.

Achondrites, unlike chondrites, underwent some change on the parent asteroid. Usually, they are volcanic magma that cooled. The melting of the rock destroyed most, if not all, of the chondrules and much of the iron migrated elsewhere. If "chondrites" means "with chondrules", then"achondrites" means "without chondrules".

Stony Irons formed deeper in an asteroid at the boundary between the mantle and the molten core of an asteroid. Iron, as it melted, migrated to the center of the asteroid. At the boundary, some iron had other minerals suspended in it. Pallasites are the most common form of stony irons and are some of the most beautiful meteorites.

Irons are formed in the molten core of an asteroid. Far from being simple hunks of metal, these meteorites can look very different when cut, polished, and etched with acid.  Irons can show Widmenstatten lines (caused when iron solidifies slowly into crystals) or Neumann lines (stress lines caused by an impact on the parent body). This is related to different nickel / iron ratios in the material.

Non-asteroid meteorites mean just that - they came from somewhere else. Some meteorites come from the Moon, and some come from Mars. We can tell this by matching chemical analyses of gases trapped in the asteroids with spectroscopic analyses of the Moon and Mars through telescopes. A major meteorite impact in either of those two places generated enough energy to send some rocks flying into space, escaping gravity and eventually finding their way to Earth.

Non-meteorite objects can be confused with meteorites, but are actually formed from Earth material melting or otherwise transforming during a meteorite impact.