Taxonomy is the science or practice of categorising things in an orderly fashion. That usually means hierarchically, so that a top-level grouping might contain several smaller groups, each of which contains several sub-groups. There's a great economy to this approach, and it meshes well with the nature of many of the things we might need to classify, notably life. The term was first used in 1735 by the great taxonomist Carolus Linnaeus in his grand work on classification, 'Systema Naturæ'.
This kind of hierarchical categorisation results in a tree-like structure, with limbs, branches, twigs and leaves spreading out from a single point of origin. That's handy, because if you know what high-level category something fits into, you might already know a lot about it - if I tell you I found a Devil's coach-horse last night, that really won't tell you much at all unless you happen to be familiar with such beasts. If I tell you that it's a kind of insect, you should immediately know it has six legs and an exoskeleton, and you might guess (correctly) that it probably has wings. If I tell you it's a beetle, you'd probably also have some idea of its body shape, and maybe you would infer that it has hardened wing-covers.
There are many pit-falls to describing things this way, though. Perhaps most importantly, the general rules describing members of a high-level category often have many exceptions, and it is not always easy to know whether to assume a tendency is shared by any given member of a class. It's tempting to say that mammals are hairy creatures which give birth to live young, for example, but platypuses lay eggs and many adult whales are hairless. There may also be things that we would like to place in more than one higher-level class, which some kinds of taxonomical approach make impossible; among living things, it is not unheard-of for individuals from two evolutionary branches to cross-breed and produce something new. Such hybrids are more common in the broader sense of taxonomy - there are a number of sciences which overlap several larger fields of study, for example, like physical chemistry, and any categorisation of cooking styles is going to need to account for multiple geographical origins.
There are several variations on the details of biological taxonomy, but the classic hierarchy goes as follows. A mnemonic for this, should you need one, is 'King Philip Calls Out For Good Soup'. You could also have 'Kangaroos Prefer Chilled Outback For Grooving Soulfully'.The Five Kingdoms are Monera, Fungi, Protista, Plantae and Animalia. chordata, along with all the other vertebrates, plus hagfish, lampreys and a couple of other sort-of-fishes. Mammalia is one of the five traditional classes of vertebrates. primates; cats, dogs, bears, mustelids and hundreds of other land mammals are in Carnivora, although a few of them, like the giant panda, are mostly herbivorous. hominidae, along with all the other great apes. Most family names end in -ae, often attached to the name of an emblematic order within that family - Corvidae, for example, includes the crows and other birds of the genus Corvus, but also magpies, nutcrackers and so on. Similarly Brassicaceae includes the true brassicas like cabbage and turnip, but also rocket (arugula) and thousands of other plants distributed among hundreds of other genera. Homo, of which we are the only extant species. All the other hominins, like the neanderthals, australopithecus, Homo floresiensis and so on are dead now, although we may have inherited some of their genes. The genus gives the first, capitalised part of the Latin name of a species, which is followed by the specific species name and subspecies if necessary, which should be written in lower-case. The whole name is conventionally written in Italics. liger offspring turn out to be fertile, and there are many cases of supposedly distinct species interbreeding in nature to produce fertile offspring. It is not always clear whether two species are capable of interbreeding, if only they had the chance and they got in the mood. One fun difficulty with the idea of a species is a ring species1, where two related animals might not be able to interbreed, but they could breed quite successfully with intermediate animals who breed with each other. The definition is also quite useless for bacteria and other life-forms that only reproduce asexually. Tea can be either Camellia sinensis var. sinensis or Camellia sinensis var. assamica, for example. There are many other ways to describe sub-groupings below the subspecies level - forms, subforms or cultivars in botany, breeds of domesticated animals, and so on. We also talk about races of humans, of course, but the idea has little scientific validity.
Some other ways of categorising
Taxonomic ranks not mentioned above include subfamily, tribe, subtribe, infraorder, infrafamily, infraclass, superorder, superfamily, subphylum and various others - perhaps most importantly, the three 'empires' imposed above the kingdoms, namely the Eukaryotes (anything with a cell nucleus), Prokaryotes (bacteria) and Archaea (ancient micro-organisms once classed with the bacteria, but which turned out to be far too weird for that to make sense). Apparently the extra levels are usually the result of trying to minimise the disruption to existing classifications when a recognised grouping needs to be moved up or down a level. Also, botanists prefer to talk about divisions rather than phyla. I can only assume that most of this stuff makes some sense to someone.
There is some debate about whether biological taxonomy should always be strictly phylogenetic - that is, classifications should always assume the existence of shared ancestors, an approach known as cladistics. For the most part, evolutionary trees fit in well with trees of characteristics, but there have been a large number of cases where groupings of organisms based on shared characteristics turned out not to be closely related at all - such groupings are said not to be monophyletic, if they include organisms from a separate evolutionary branch. For example, although the butterflies are monophyletic, the moths are not - while we might be able to make some generalisations about moths, they are really just those members of Lepidoptera that are not butterflies2. On a much grander scale, the reptiles turn out to be a surprisingly messy grouping; in order to use only strictly monophyletic categories, we would need to count birds (and hence dinosaurs) as reptiles3, or else put crocodilians and birds in another class together, since they are more closely related to each other than they are to the likes of lizards. We might also divorce the turtles and a few others from the rest of the reptiles. This is all very inconvenient, given how many textbooks and so on have been written on the assumption that it is legitimate to talk about 'reptiles', and this question is not likely to be settled for a while yet.
In the computing world, tags are sometimes used instead of, or alongside taxonomies. The main difference is that tags are not usually hierarchical in any sense. Relatedly, the only limit on what tags something can have is usually numerical - you might be restricted to ten tags per photo, for example, but in purely taxonomic classification you can usually only belong to one of each level of category (one phylum, one order, one species and so on). Tags tend to be very informal, with users allowed to tag their contributions with whatever they fancy, but they are sometimes used very specifically so that computers can make sense of them.
In many contexts, it probably makes sense to give up on categorisation altogether and just think of things in terms of adjectives. This may be at the expense of being able to make inferences which are likely to be broadly correct in any particular case based on only a few words, but it has the considerable advantage of acknowledging how many things (and people) don't fit neatly in any of the available categories, and avoiding the misconceptions that arise from expecting them to.
References1Mark Ridley's Evolution on ring species
2Bug Guide on order Lepidoptera
3Tree of Life on Amniota