Livescience sums up the main various classifications of heavenly bodies in the solar system with an article by Tariq Malik.
Planets are the (now) eight large bodies orbiting the Sun, having enough mass to be 'nearly' round, and having cleared its surrounding orbital path of other solar objects. The caveat surrounding the object formerly known as the planet Pluto is that last little requirement. However, some astronomers still object, as the author notes:
"Pluto, while round and orbiting the sun, is one of a swarm of so-called trans-Neptunian objects, small icy bodies in the comet reservoir of the Kuiper Belt that extends out from Neptune’s orbit, leading to its IAU demotion. But critics have said that asteroids can be found accompanying established planets like Earth, Mars and Jupiter, throwing a wrench in that requirement."
Dwarf planets are objects that meet just the first two criteria, meaning the object formerly known as the asteroid Ceres meets the new definiton, while the other object what were known as asteroids are now 'solar system bodies' which I find a rather inelegant and stupid term - at least compared to 'asteroids'. Some astronomers still call these object 'minor' planets, which was an earlier (if I recall correctly) term used for space rocks orbiting the Sun.
Then there are those dwarf planets that are also located out beyond Neptune (trans-Neptunian objects, TNOs) called 'plutoids'. This classification was created just this summer, apparently as a sop to those of us upset with the earlier demotion of Pluto. There are also a number of other 'non-official' names for some of these very distant objects, such as KBO (Kuiper Belt Objects) or EKO (Edgeworth-Kuiper Objects) both names referring to the same objects as TNOs but honoring astronomers theorizing the existence of such objects in the middle part of the last century before their actual discovery near the end of it.
Then there is the name 'plutino' which is an object which lies in orbital resonance with the planet Neptune - rather than Pluto, which is actually the first such object of which we were aware. Such objects orbit the Sun three times for every two orbits of Neptune, thus the classification refers to its orbit and not any physical characteristic.
And all this does is brush the subject - there are also cubewanos (objects near Pluto that aren't plutinos), Trojans (objects sort of like plutinos only orbiting in front of or behind the oribital path of Jupiter), SDOs (scattered disk objects, or those that aren't in the normal orbital plane that the planets and most objects circle the Sun), NEOs (near Earth asteroids), PHAs (potentially hazardous asteroids which could hit the Earth someday), and OCOs (Oort Cloud objects such as comets, and even further away than the TNO/KBOs).
More explanation on the more obscure of these terms are explained at Space.com