The concepts of monophyly and paraphyly are at the very heart of modern efforts to describe the evolutionary tree of life. Don't let the number of syllables scare you away; they're actually really simple, but may take some time to "sink in". Your efforts will be rewarded handsomely; see how you impress your dinner guests when you casually drop "paraphyletic" in conversation!
Loosely speaking, they just describe how "natural" a group of species is. Monophyletic groups are the "best": they contain one species and all its descendants. If any descendants are missing, they're called paraphyletic. If the members are unrelated (i.e. if a group lacks a common ancestor), they're called polyphyletic.
Consider this phylogenetic tree of Reptilia: (Note, reptile phylogeny is still being hotly debated!)
The group most people refer to when they say "dinosaurs" is actually not completely natural it is paraphyletic because it is missing birds. (Yes, birds are direct descendants of dinosaurs!) However, lizards and snakes together form a nice neat little monophyletic group, the Squamata it contains all descendants of that first proto-lizard and nothing else. Warm-blooded creatures, on the other hand, are a polyphyletic group mammals and birds, for example, while both warm-blooded, are unrelated. (Their nearest common ancestor was almost certainly cold-blooded.)