Linnaean taxonomy, which is the dominant system in use for over 200 years, allows groups that are paraphyletc or polyphyletic.
A polyphyletic group contains members that have a common ancestor but exclude many descendents of that common ancestor. For instance, the group of warm blooded Tetrapods was once called Haemothermia, including the birds and mammals and any extinct warm-blooded dinosaurs etc. This might be a useful category for some purposes, but not for the purposes of taxonomy, and bit by bit people have been weeding these cases out of the Linnaean system. Birds are much more closely related to crocodiles than they are to mammals, so under the Phylocode system, such a grouping would be not allowed: any group that contains birds and mammals should also contain any beasts more closely related to birds than to mammals … and also those more closely related to mammals than to birds.
A paraphyletic group contains some but not all of the descendants of a particular group. An example would be Reptilia. Reptilia contains all of the descendants of, say, the earliest Amniotes … except for Mammals and Birds. This too would be not allowed under Phylocode. If you are including the earliest Amniotes then you must include their descendants, the Mammals. And if you’re including Turtles and Crocodiles, then you should also include Birds, since Crocodiles are more closely related to Birds than they are to Turtles.
Phylocode permits only three kinds of clade definitions:
Stem based: “the clade consisting of A and all organisms or species that share a more recent common ancestor with A than with Z”
Example: the clade that consists of humans and all species that share a more recent common ancestor with humans than with orangutans
Derived character (apomorphy): “the clade originating with the first organism or species to possess characteristic M inherited by clade A”.
Example: the clade that originates with the first species to have pennaceous feathers ultimately inherited by modern birds.
Node based: “the clade originating with the most recent common ancestor of A and B”
Example: the clade originating with the most recent common ancestor of humans and pigeons
The last two hundred years have seen huge shifts in taxonomy and phylogeny, as different kinds of morphological and fossil evidence have come to light and been considered. We are now in an era where we can think that the shifting will come to an end as genome comparison becomes more absolute.