The first pteridophyte is one of the commonest in this country: hart's tongue fern or Asplenium scolopendrium. This one is in the garden providing a magnificent green luminosity to one of the darker corners.
Just to get the plant kingdom a bit clearer in my head: it is divided into green algae and the land plants (Embryophytes). The latter are subdivided into the non-vascular land plants (Bryophytes - essentially the mosses, hornworts and liverworts); vascular plants (Tracheophytes - mostly ferns and horsetails (Pteridophyta) and clubmosses); and seed plants (Spermatophytes - mostly the conifers (Pinophyta) and flowering plants (Magnoliophyta)). So hart's tongue is a pteridophyte (Division), and in this, it belongs to the Polypoliopsida (Class). Its order is the Polypodiales which contains around 80% of ferns. The genus belongs to the family Aspleniaceae, the only other genus being the Hymenasplenium. There are around 700 species and they are often referred to by their common name: the spleenworts (from the doctrine of signatures which held that the sori looked somewhat spleenish! (wort is just an old English word for plant.))
Apparently the pattern of the sori is reminiscent of centipede's legs and the name scolopendrium derives from the Latin for centipede - weird!