Carl von Linne, a Swedish botanist (plant scientist) known as Carolus Linnaeus (Latin was the common language for European science, so writings and often names were Latinized) began work in 1735 on a system that would organize descriptive classification from the smallest of related groups up to the very largest.  The system he and others developed, with revisions, is the basic system still commonly used today to systematically organize types of living things with their relatives.  The basic structure was similar to how human organizations work, with groups-contained-within-groups, be they feudal power structures or military organizations.  Each particular type of living thing would be a species (from the same root word as "specific").  Closely-related species could be collected within a larger grouping, a genus;  related genera are grouped into a family, families into an order, orders into a class, classes into a phylum, and phyla into a Kingdom, the biggest and most general group.  Today, Kingdoms are in Domains.  In Linnaeus' time, there were just the Animal Kingdom and the Plant Kingdom, but later discoveries convinced biologists (then called naturalists) that some distinct types of organisms, such as Fungi and some tiny single-celled organisms, should be given their own separate Kingdoms.

Some subdisciplines of biology use a basic Linnaean type of taxonomy, but may change the basic names used for a few of the groups.  Commonly, for instance, plant and fungus taxonomy uses the term Division instead of Phylum.  How groups are defined has changed as well, with clades indicating shared key characteristics and ancestry. Recently, there has been a bit of a movement to revamp the basic system to something called phylocode.

As you might see from the examples above, the system is a little more complicated than it first sounds.  Sometimes, two or more groups are found to be more closely related than anyone thought;  they might be connected as supergroups ("super-" put on one of the level-groups name).  And sometimes a group is not as unified as was thought, and is split into subgroups.

Biology, like the sciences in general, is produced by human beings who often disagree with each other's ideas and fight over what the "proper" labels should be.  For the most part, it is perfectly allowable for someone to say that this or that group should be a subphylum rather than a phylum, or a family rather than an order (this will come up later in the discussion about Kingdoms).  However, what is not allowed is to, on a whim, change the name of a particular group - you can't say, "I dont like the genus Ursus for bears, I and everyone I work with are going to use Yogi from now on."  Once a group is named and the name accepted, it may be tossed up or down the "classification ladder," but one must gain a broad consensus and acceptance before a group's actual name is changed.  If one book places sponges in their own Kingdom and one puts them in a phylum, in both cases the group will have the name Porifera;  this limits confusion when doing background research on organisms.

Another set of rules, called binomial nomenclature (2-name naming), determines how species names are used.   You'll see in the examples above that species names are two words:  a capitalized genus name and an uncapitalized specific.  The second word has no meaning by itself, and is never capitalized, not even if a proper noun is used as the source of the term.  Species names (and Genus names) are also treated as foreign words in English, meaning that they are italicized or underlined when printed or written.  The names of other taxonomic groups may also be italicized or underlined, but that usage seems to vary.

Typically, species names are abbreviated by making an initial of the first word and spelling out the second - you may be familiar with E. coli,  the abbreviated name of Escherichia coli, a common intestinal bacterium that, if introduced into an incompatible intestine, can cause food poisoning.

Deciding what living things should be classified together in the same group requires deciding what's related to what, and how close those relationships are.  Long ago, it was often done by lumping together analogous traits:  features used to do the same function.  This is why, in Biblical times, if they had finny structures and swam ("Beasts of the Water"), or wings and flew ("Creatures of the Air"), they belonged in the same groups.  By this approach, long wiggly things like snakes were grouped with earthworms and eels.

As more and more people studied Nature in detail, it became obvious that a butterfly's wings were very different structures than a bird's wings.  And sometimes, it could be seen that two structures used for very different functions - such as a human hand, a bat's wing and a whale's flipper - all contained the same internal architecture, with sometimes subtle changes in internal parts producing the outward changes.  Traits with similar internal structure are called homologous traits, and it was eventually decided that these traits were a better measure of relatedness than analogous traits.  Keep in mind, however, that traits can be both analogous and homologous (like a monkey hand and a human hand), it isn't automatically an either / or situation.

The modern approach to classification is very focused on critical traits that arise and characterize a new family line - histories are based upon the spot in the past when a particular trait arises.  This approach is called cladistics.

There is currently a strong shift toward using DNA comparisons as the main discriminator in taxonomy, but deciding just how much difference is enough to set group separation is very much a work in progress.

Much basic taxonomy is still done anatomically, although the details used have gotten smaller through the use of microscopes and broader through the discoveries of genetics and biochemistry (yes, molecules have a sort of anatomy - the homology is in the sequence order of the components). 

The original two Kingdoms were Plantae and Animalia (and the minerals, but they went off on their own), which remained the only Kingdoms until the middle of the 20th Century.  During the last 40-50 years, those groups have been splintered a bit - Fungi was split off from the plants, Protista removed the single-celled eukaryotes (and the problems of their often-combined characteristics) and Monera was made for the prokaryotes.  Those five Kingdoms were considered "the" Kingdoms in most basic biology books, even though the splintering has continued.  The latest basic books now recognize a sixth group, the Archaea, once thought to be odd monera / bacteria but now considered a fundamentally different group. 

As mentioned, most modern systems (mostly in the plant and microbe areas) add a Domain level above the Kingdoms;  most commonly, there are three domains.  The Monera and Archaea (seen below as Kingdoms) are each Domains, with the rest of the Kingdoms in the Eukaryota Domain.

The modern definition of classification groups depends upon each species in the group evolving from a single ancestral type with the basic group characteristics - plants all share an ancestor that had true plant characteristics, but the ancestor they share with fungi is neither distinctly plant or fungus, so they have been designated into different Kingdoms.  Based upon this criteria, many zoologists think that the Animal Kingdom should be splintered into at least two Kingdoms.  The Protista and the Monera are often "made up" of multiple Kingdoms in advanced books on the subjects.

By this definition, any group is supposed to be monophyletic, where every member can be traced from a single ancestor that can be included in that group.  If this is not true, a group is said to be polyphyletic, which is a criticism;  it implies that multiple groups that shouldn't be lumped together are being classified incorrectly.

Keep in mind, like all aspects of classification, this fits into the convenience of human labelling, which doesn't always comfortably fit what the real organisms are doing.  And with that fresh in mind...

Labels require definitions, and species is a very particular label that has been defined in different ways through time.  It first just meant a distinctly-describable type;  then it was distinct types that could not interbreed;  then it was distinct types that could breed and produce offspring that themselves could go on as adults to breed (some crosses between species can produce young, such as horses and donkeys producing mules, but they grow up to be sterile adults).  Today, the best, latest nontechnical definition of species is...

Species:  A group that, in natural surroundings, breeds exclusively within the group.

In effect, we now let the organisms themselves determine what belongs to their species and what doesn't.  This still is not a great definition - it says nothing about asexual species.  And, like almost any biological definition, it still has exceptions, such as with coyotes, dogs, and wolves.  But it works fairly well.

Before long, there may be a strong attempt to define species genetically based upon molecular differences.  This sounds simple and mathematical, but it isnt;  don't expect a reliable standard any time soon.

A book from 1866 on the classification of animals.

Lists of odd species names.