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Kingdom Animalia
Phylum Chordata
Class Mammalia
Order Primates
Family Hominidae
Genus Homo
Species sapiens

The above is the Linnean classification for humans.
Species is defined, but the concept is confused in assexual reproduction, since one of the key criteria is inability to reproduce (or unwillingness to reproduce in nature). There are very slight differences between a population where one species arose and the daughter population. And sometimes even sexual reproducing organisms can reunite and produce a new species with combined characteristics. (This is why Darwin's finches are so confusing to creationists.)
[The biological species concept is based on the idea that a species is a group of interbreeding individuals who are reproductively isolated from other species. Reproductive isolation can be genetic (the hybrids are sterile), behavioural (individuals of the two species are not able to mate because of differences in courtship, for example) or physical (they are separated by geography). This concept is of course not useful for fossils. Species boundaries for extinct species must be created cautiously since this type of data is unavailable. ]

The above is the reason that fossils are difficult to seperate as species. No biologist has observed their behavior directly, so they look at behavior in existing species and combine them with forensic information to arrive at conclusions, like whether something is carnivorous or herbivorous by adaptation of teeth.

Admittedly, the judgement on what is a new species must contain some subjective judgements based on experience. On the other hand, some species are different based on totally objective criteria that the two populations do not interbreed.

These are descriptions of different types of speciation. This does not explain the mechanism of speciation, but is simply description of an observed pattern among various populations under study;

Allopatric speciation: (allopatric means literally "different homelands") in this model two populations which are geographically isolated from each other (eliminating the possibility of gene flow) can gradually diverge to form two distinct species.
Sympatric speciation: (sympatric means "same homeland") species arise in this model despite being in the same location. Some other effect must isolate a subpopulation from the rest of the population; this is often behavioral, and often occurs very rapidly.
Parapatric speciation: (parapatric means "near the homeland") species can form at the border between two populations. Often this border mirrors an actual geographic border, and the species formed occurs at this margin.

All of the larger groups above species are rather subjective, and are subject to great debate among taxonomists on the fringes, where transitional organisms lie. (When I say transitional, this does not mean that they are at the node of a separation of species, but have continued to evolve up a branch, but still retain many characteristics of two sets of larger classifications.

Currently cladistics, a whole different way of classifying by related groupings is popular. I am not thrilled with it, but it seems to help in certain fields like paleontology.