Taxonomy is the study of the degree of biological similarity of organisms; the end product of a taxonomist's work is generally an arrangement of taxa in such a way that the taxa which are most biologically similar are grouped closest together in the arrangement, and the relative closeness of taxa in the arrangement reflects their degree of biological similarity in real life. The theoretical categories that taxonomists use as place-holders in such an arrangement ( species, kingdoms, etc.) are known as taxa (singular taxon)
Exactly what type of biological similarity is most relevant is the sort of thing that taxonomists spend their time arguing about. Most of the time, they use the term relationship where I have used the term biological similarity, ignoring such things as predator-prey relationships, geographic relationships, and a host of others. But using this (" relationship") terminology, we would rephrase my above definition as " taxonomy attempts to discover the degree of relatedness between organisms."
Nowadays, the relevant type of biological similarity (or relationship) is usually taken to be evolutionary distance, that is, the amount of biological change or (estimated) length of time that two taxa have piled up since diverging (through evolution) from a (usually theoretical) common ancestor. This distance is measured through mating studies (when feasible) and DNA analysis.
Before (1) the acceptance of Darwin's theory of evolution and (2) the acceptance of sexual reproduction in fungi, it's hard to conceptualize exactly what sort of biological similarity mycological taxonomists were arguing about. Ernst Mayr has written that the old concept of taxa was that of a "class", a rather Platonic notion that united its members by their joint partaking of a sort of idealized essence or notion. For instance, "white-spored, ring on stalk but no volva = Lepiota". This was certainly Linnaeus' notion, and seems to have been Fries' as well: he spent his life searching for the ideal characteristics that would magically sort the fungi into clear and logical groups. He got a quick start with spore color, but the quest quickly bogged down after that. There simply isn't any such character: different characters are useful in separating the members of different groups, and at different levels. Features that are useful and constant in some groups are very variable in others. The fungi just didn't develop with out platonism in mind.
This didn't prevent 19th century mycologists from pursuing their Platonism, of course, so they got into big arguments over whether a given fungus belonged in one idealized group or another. Most of those arguments now look very silly, which has made mycologists much more circumspect in their arguments today.
I hold the dominant view in biology today, that a species is an interbreeding population (or group of such populations) that is unable to breed outside the group without clear genetic penalties (such as sterility). This is known as the biological definition of a species, since it emphasizes the biological compatibility of the interbreeding population. I emphasize that I consider the species to be something real, that occurs in nature. In the first paragraph of this entry, I referred to the species as one of the "theoretical" categories that make up taxonomies. This "theoretical" only refers to the fact that we don't actually physically take the real organisms and somehow incorporate them into taxonomies - - instead, we make a description of some sort that stands in for the genuine species.
The next "main" taxon above the level of species is the genus. This is the most basic grouping of species for the animal and plant kingdoms. For example, the genus of cats, Felix, contains the species Felix leo, F. tigris, and F. domestica. In fungi, it's a little more complicated because many genera are so huge (2000+ species of Cortinarius and counting) that they are divided into sections (and often into subsections, stirps, and so on).
The sections of most genera have, at one time or another, been "raised" in status to independent genera, causing much complaining that this gives us a new name to learn. Such complaints miss the mark. Take, for example, the genus Ganoderma and its division into sections Ganoderma (for shiny-capped ganodermas like Ganoderma lucidum) and Elfvingia (for non-shiny ones, like Ganoderma applanatum).
If it's helpful to know these two taxa as two distinct groups, each with its own name and set of characters, then it doesn't really matter if you call them two sections or two genera. All that's at stake is the "status" of the genus, and I find it hard to believe that any of these taxa care about their taxonomic status. Similarly, many recent papers have raised traditional families of mushrooms (for example, the Tricholomataceae) to the rank of order (Tricholomatales). This change in rank is meaningless in and of itself - - there are no requirements or essential characteristics for a taxon to be an order rather than a family, or a genus rather than a section. But the change in ending is helpful when you are discussing your new taxonomic system, as it makes it clearer when you are talking about the "old" Tricholomataceae and when you are talking about your new, improved version. This is a pragmatics issue, rather than a scientific one.