Monophyletic: a group of taxa comprised of a shared common ancestor and all of its descendants.
Paraphyletic: a group of taxa comprised of a shared common ancestor but NOT all its descendants.
Polyphyletic: a group of taxa which does not also include their common ancestor.
For example, take the tree: mammals + (turtles + (lepidosaurs + (crocodilians + birds)))
"Reptlia" as traditionally concieved is turtles, lepidosaurs, and crocodilians. Their shared common ancestor is also considered a reptile. However, this grouping excludes birds (because they are so radically transformed). Thus, the traditional "Reptilia" is paraphyletic.
Reptilia as used by Gauthier (and many others now!) comprises turtles, lepidosaurs, crocodilians, birds, their common ancestor, and all of its descendants. This grouping is a monophyletic grouping.
Some workers have proposed a taxon "Haematothermia" (the warm-bloods) for birds plus mammals, but nothing else. Given the tree above, the common ancestor of the haematotherms would NOT also be a haematotherm, and so Haematothermia is polyphyletic.
Hope this helps.
Thomas R. Holtz,