Phylogenetics is the study of organism “relatedness.” Attempting to sort plants phylogenetically is similar to you sketching your family tree. It's often obvious from looking in the mirror or from studying the diagram that you are more closely related to a sibling than to a great aunt. But when you get into your relationship to a third cousin vs. a first-cousin-twice-removed, things can get a little fuzzy.
Richard M. Smith, in explaining the arrangement of plants in his book Wildflowers of the Southern Mountains, said:
The first group of families, from the Typhaceae through the Orchidaceae, consists of the monocotyledons, which typically have parallel-veined leaves and 3- or 6-parted flowers.
The remainder, which are the dicotyledons, are characterized by net-veined leaves and mostly 4- or 5-parted flowers.
The Botanical Gardens at Asheville has this to say:
The families are arranged in a standard phylogenetic arrangement, i.e., from “primitive” to “advanced”....
Ferns, being simple non-flowering plants, are placed first....
The water plantain family (Alismataceae) is considered relatively primitive, while the orchid family (Orchidaceae) is considered the most advanced family of monocots.
Of dicots, the aster family (Asteraceae) is most advanced in its floral characteristics and is placed at the end of the list.
Today's molecular research is revealing previously unknown relationships, and some traditionally accepted groupings are being rethought: Be on the lookout for terms such as “primitive Angiosperm,” “Magnoliid,” and “Eudicot” to start entering the conversation.
In the meantime, when this website sorts “phylogenetically,” the sort order is based on “index numbers” found in the Vascular Flora of the Carolinas, which assigned each plant a number designating its position on the family tree as it was understood in the 1960s.