There are bodies that govern this but it's quite confusing. Wikipedia has several methods to deal with this but there are a lot of historical definitions, items that are missing and of course debate.
I've organized everything into my own schema. Objects are classified by type and then by size:
Planets (our solar system only) are further divided:
Outside the solar system a few new planet types have been discovered. Hot Jupiters, super Earths, mini Neptunes, etc. It is probably too soon to propose a comprehensive schema for planets.
There are many, many groups of asteroids. Some are listed here:
It should be noted that asteroids are primarily classified by location rather than type, composition, chemistry, etc. These probably do not belong in my schema at all.
Also it should be noted that the difference between asteroids and comets is now known to be blurry and really they should be considered the extreme ends of a spectrum (from pure rock to pure ice).
Lastly, distant minor planets:
I think there's a ways to go before we have a well organized classification. If we even need one. Multiple, overlapping definitions has its advantage too. But so does common terms we can use in writing and communication.