Mayflies are one of the most basal, if not the most basal groups of winged insects alive today. The one thing that everyone knows about mayflies is that they only live for a day. This is reflected in their order name Ephemeroptera - "fleeting wing". But like so many other things that "everyone knows", this is complete twaddle. Mayflies do not live for a day. Indeed, by insect standards mayflies often have very long lives - many will live for an entire year, and some will live for as long as three years. The "fleeting" part of their lives is not their life as a whole, but only their time as adults.
Like dragonflies, mayflies spend the juvenile part of their life cycles as aquatic nymphs. They are easily distinguished from the aquatic nymphs of other insect orders by their three long caudal filaments and the leaf-like lateral gills that run down each side of their abdomens. Most mayfly nymphs are herbivorous or detritivorous, but a few are predatory. In contrast, adult mayflies have almost non-existent mouthparts, and do not feed - hence their "brief" lives. Mayflies are also unique in being the only living winged insects to undergo a moult after their wings develop. They first emerge from the water in a form known as the subimago, which soon moults into the final, fully mature imago. Mating usually happens in the imago stage, but there are some species that find themselves unable to wait that long, and begin mating as subimagoes (Grimaldi & Engel, 2005).
The contrast between mayflies' long juvenile lives and brief adult lives highlights a common prejudice to which almost all of us are prone - we tend to think of most animals in terms of their adult forms rather than their juvenile ones. There are a number of reasons for this. One is that we ourselves spend more of our life cycle (if we're lucky) as adults than juveniles. Another is that the adults are usually larger and more visible than the often reclusive juveniles, so we're more likely to notice them. Even working biologists may be prone to this trap - many animals can only be fully identified once they reach adulthood, which leads to the subconcious tendency to dismiss the unidentifiable juveniles as unimportant. But from an ecological perspective, it is the long-term juvenile mayflies that are far more significant than the brief adults. Adulthood in mayflies is simply a coda, a brief interlude to prepare for the next movement. Similarly, one could argue about how appropriate it is to characterise mayflies as flying insects when for so much of their lives they are not, with flight for them having no other purpose than to facilitate the reproductive process.
An influential classification of recent mayflies by McCafferty (1991) divided them into three suborders, Rectracheata, Setisura and Pisciforma, largely on the basis of their tracheal anatomy. Unfortunately, McCafferty did not explicitly refer to the characters on which he based the Pisciforma, other than noting that the name referred to the "minnow-like bodies and actions of the larvae". McCafferty (1991) regarded the Pisciforma and Setisura as forming a clade to the exclusion of the Rectracheata. In contrast, Kluge (2004) used an alternative classification* that more or less united McCafferty's Setisura and Rectracheata into a clade Bidentisetata, with two tooth-like setae on the maxilla (I say more or less because the families involved were not exactly the same), separate from the "Tridentisetata" (effectively McCafferty's original Pisciforma) with three tooth-like setae. Kluge explicitly stated, however, that his Tridentisetata was a taxon of convenience united by plesiomorphies only and probably paraphyletic with regard to the Bidentisetata. Recent molecular analyses have confirmed that the "Pisciforma" are paraphyletic or polyphyletic. Ogden & Whiting (2005) found both Rectracheata and Setisura nested within Pisciforma, and suggested that the fish-like nymphal form was plesiomorphic for all living mayflies and lost on numerous subsequent occasions.
*Very alternative, in fact. Kluge (2004) introduced a new system of nomenclature that attempts to provide an alternative to the rank-based system. Without wanting to go into too many details (for a start, that would require me to actually follow what's going on there, and frankly I haven't got a clue), Kluge's system involves a combination of a type genus plus a suffix indicating a taxon's position in the taxonomic hierarchy. So Arthropoda, for instance, is referred to by Kluge as Araneus/fg7. Seriously.
There is one little detail that I have to mention before ending this post. One characteristic of the wings of living mayflies as a whole is a significant difference in size between the fore- and hindwings, with the forewings significantly larger. This is taken to its extreme in one of the pisciform families, the Baetidae, which are one of the few groups of insects in which the hindwings have been almost entirely lost, reduced to small pair of buds like those of flies.
Grimaldi, D., & M. S. Engel. 2005. Evolution of the Insects. Cambridge University Press: New York.
Kluge, N. 2004. The Phylogenetic System of Ephemeroptera. Springer.
McCafferty, W. P. 1991. Toward a phylogenetic classification of the Ephemeroptera (Insecta): a commentary on systematics. Annals of the Entomological Society of America 84 (4): 343-360.
Ogden, T. H., & M. F. Whiting. 2005. Phylogeny of Ephemeroptera (mayXies) based on molecular evidence. Molecular Phylogenetics and Evolution 37: 625-643.