The question seems simple enough. But how would you describe a "species"? How would you describe the difference beween human beings and cats, for instance?
Are species defined as organisms with distinctive characteristics that distinguish them from other species? (morphological)
Are species groups of organisms that reproduce with one another? (biological)
Are species defined by their ancestors? (Darwinian)
Are species determined by the organisms that individual members would choose as mates? (mate-recognition)

Given our prior discussion on taxonomy, you can see how the ambiguity in definitions can cause some problems. Biologists have long debated which species concept accurately describes the notion of a species. Clearly the biological and mate-recognition approaches seem wrong because they exclude organisms that produce asexually. But this still doesn't leave us with a clear definition of species.
The problem seems to be that different people want to answer different questions.
What are species?
How do we identify species?
Why are there species?
What factors are important in speciation, the creation of new species?
Certain species concepts seem inappropriate for discussion of some of these issues. Disagreement on the answers to these questions make the issue that much more problematic.
What are species?
Many biologists in the phylogenetics realm (that's me for the purpose of this discussion) currently adhere to the Darwinian (or evolutionary) species concept: species are lineages of ancestor-dependent populations. Relationships between species form a network of "reticulations". A species starts when an organism diverges from its lineage, and ends at extinction or the creation of new daughter lineages.
We've seen trees in the taxonomy article. Imagine the "Tree of Life", the tree that describes the relationship between all species. The tree is a hierarchy of historical lineages and current species.
Construction of the Tree of Life is a fascinating and enormous task. There are people who devote a great deal of time to building this tree. The Tree of Life Project website is a great resource if you're curious about how far along we are in understanding the evolutionary history of all organisms. The site includes an impressive guide to the tree, along with pretty pictures and extensive discussions.
How do we identify species?
Clearly we want to identify species by their character evidence. In terms of phylogeny, we identify character traits unique to lineages. It's a fuzzy description: how do we determine the cut-off? Often times we define "clusters" of characteristics as species descriptors.
Why are there species?
This is like asking "why do we evolve"? Natural selection and adaptation would be my answer. It turns out there are a few theories, some completely bonko:
Typological species: species are creations rather than evolved entities. (Move aside, god-people, that's just rediculous!)
Biological species: species are maintained by reproductive boundaries; a new species is created when the boundary is crossed (e.g. a horse mates with a donkey).
Ecological species: species are groups of organisms that occupy specific ecological niches. (this doesn't seem to tell the whole picture to me)
Cohesion species: emphasizes cohesion mechanisms that hold species together like reproduction and demographics.
Some hippies argue that species don't really exist except for in the minds of the biologists. Please. Because there is language, there is an inherent necessity to define and categorize everything. So in that case, nothing exists except for in our own minds. Like, dude, like, maybe we don't really exist except for in the minds of some bigger super-dude.
Are you a living computer simulation?
A philosopher's playground
You may have noticed that organism distinction is as much a philosophical issue as it is a biological issue. If you're at all like me, you tend to veer from the latter topic, as it can be most excruciating and (this is coming from someone who doesn't like philosophy) usually a waste of time. Still, even for me, the topic is interesting in its inherent social dilemma. Do categories really exist in nature? If you're at all interested in this, here's an article on the philosophy of species concepts with some references.
What mechanisms are important in speciation?
According to different species concepts, different mechanisms are more important than others: mates, physical characteristics, evolutionary history, ecological demographic. Are these characteristics at all compatible so we can at least define the concept of a speciation event?
The evolutionary definition is defined by the evolution of characters. Often new lineages arise as a result of ecological or reproductive change. So perhaps we can bring it all together at the speciation level. Let's consider what has to happen in order for speciation to occur.
Why can't we have sex?
Let's think about the reproductive aspect. We have some vague idea that many new species come about as a result of reproduction. So what keeps this from happening?
Prezygotic Isolating Mechanisms:
-Incompatible or absent mating calls.
-Different sub-habitats of the same habitat.
-Incompatible sex organs.
-Different breeding times.
-The gametes don't combine.
Postzygotic:
The species might reproduce, but the resulting organism has a low chance of survival. (We'll talk about hybrids soon, I promise!)
Horse + Donkey = sterile donkey = bad news for the ladies
So how does speciation occur in light of these isolating barriers?
Basically, a lineage splits when the two or more resulting species can no longer reproduce due to isolation barriers. This can occur in several ways.
Allopatric model: The formation of two of more species due to geographic separation (e.g. continental drift, a river, etc.).
Peripatric model: A small popluation of an existing population becomes isolated at the "edge".
Parapatric model: The two species exist in the same geographical area in groups that abut one another but never overlap. For instance, the dividing line may be the edge of a forest or a crossable river.
Sympatric model: Two or more dependent species evolve from the same ancestral species in the same geographic location. (Many biologists don't think this exists, as the conditions for such speciation are rare.)
Next time: Hybridization
Cute site: Evolution 101