The Palaeoptera is the group of primitive (basal) winged insects which do not fold the wings back over the abdomen, as Neoptera do. Most of them are extinct; the living ones are the Ephemeroptera (mayflies) and Odonata (dragonflies and damselflies).
The Palaeoptera may not be monophyletic: the issue is not yet settled. On the other hand, the Neoptera definitely are monophyletic.
The absence of wing-folding does not necessarily mean the Palaeoptera form a natural group. They may just be a group of insects that "are not Neoptera", in other words, a wastebasket taxon. As it stands, the relationship of the two living Paleopteran groups to the Neoptera is unclear.