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A species is a taxonomical group (specifically it is the lowest, and most precise level in the hierarchy) of organisms with similar characteristics and behaviours that are readily capable of interbreeding and producing fertile offspring. The term is somewhat loose with respect to the fact that not all people can agree when a new species has been produced by natural selection, and interbreeding is possible between different species of animals if they are closely related enough. Additionally, wildly different breeds within a species, such as with dogs, may have trouble naturally mating for practical reasons, confusing the definition of species further.
- The above definition fails for creatures which reproduce asexually, or for which the biological test "is not available".
Problems with "species"
The term "species" is malleable. The pigeon-holing of living things into one or another specific has more to do with human classificatory systems than biological reality. Humans have attempted to categorise according to "like" and "unlike" examples of things since at least the time of Aristotle and although such systems are useful for our purposes they're not necessarily a feature of the natural world (or at least not in such simplistic terms). Most of the work in classifying into species took place prior to modern genetics, and even now - with that genetic knowledge - it isn't used to thoroughly define our definitions. If anything, genetics shows that rigorous black-and-white categories like "species" aren't an especially strong idea. So any argument along the lines of speciation causes things to become different species, which means they can't reproduce, which means they'll go extinct, which means that evolution can't be true!!! (see below) is really in the formal category of not even wrong, because it tries to apply a thoroughly man-made concept to something that just happens naturally.
That isn't to say that the definition doesn't have a use. Indeed, it's the usefulness of categories and classification that led to the term being developed. But when you use them, remember that natural behavior doesn't have to conform to the categories we develop.
Definitions of "species"
One naturalist's viewpoint:
'It is perhaps appropriate to define the terms species and genus. This is not straightforward. When Darwin posed the question "What is a species?" (The Origin of Species, Chapter 2. 1859) the answer he gave was "No one definition has yet satisfied every naturalist". Our understanding of the animal kingdom has moved on since then, but almost a century and a half later the question still remains unanswered. There are a number of competing definitions, the two most frequently used being the Biological Species Concept (BSC) and the Phenological Species Concept (PSC). The problem is that, excluding the possibility of saltation (the evolution of a new species in one abrupt step), there can be no precise definition of a species. Species have to evolve over a period of time. In the case of our own species Homo sapiens, there must have been a gradual transition from our progenitor, Homo erectus (or possibly Homo ergaster, see R. Dawkins 2004. The Ancestor's Tale. London: Weidenfeld & Nicolson) that we can only identify from a historical standpoint: there was no Mitochondrial Eve from whom we are all descended. Mitochondrial Eve is shorthand for a core group, perhaps 2,000-10,000, of African hominids who lived about 150,000 BP and gave rise to the species we now describe as Homo sapiens: see S. Oppenheimer. 2003. Out of Eden. London: Constable & Robinson. The BSC defines a species as a group of organisms that freely interbreed in nature producing viable and fertile young. It suffers from the problem of what to do with groups that do not overlap in the wild making it impossible to say whether they would interbreed or not. The PSC defines the species as the smallest group for which there is parental pattern of ancestry and descent, but again this leads to subjective decisions and is not a rigorous definition. The advantage of putting things in neat boxes is too obvious to need expansion, but the problem with such boxes is that they have no reality in nature. Alfred Russel Wallace (1823-1913) recognised this dilemma when he said "there is no possible test but individual opinion to determine which of them shall be considered as species and which as varieties." (Quoted by Darwin in The Origin of Species, Chapter 2, 1859). Different species concepts suit different groups of people: evolutionary biologists may find the PSC most appropriate, taxonomists the BSC, but for naturalists and the general reader the second part of Darwin's answer "yet every naturalist knows vaguely what he means when he speaks of a species" seems to me the best pragmatic approach for naturalists and the general interested public. Let us call it the Darwinian Species Concept (DSC). [...] A genus is defined as a group of species that are more closely related to each other than to any other species." 
To summarise this extended joke: naturalists can with experience, sometimes with difficulty and argument, identify members of a species - though not as fast or as accurately as actual members of that species, especially when they're horny. 
One step "below" species is Subspecies. For a population of a species to be a subspecies it must meet two criteria. First, it must be physiologically different than the rest of the species. Second, it must be mostly genetically isolated from the rest of the species. This is generally the first step in speciation; different populations of a species will become isolated from one another, and they will begin to adapt to different environments or genetic drift occurs. Any number of things can happen from there, one of the more common being ring species where nearby populations can mate with each other but distant populations can not. If a member of a population is capable of producing a viable offspring with a member of another population but will not successfully breed without human intervention, it is considered a separate species. For example, dogs, wolves, jackals, coyotes, and dingos are almost the same species and can mate with each other (note, DO NOT PET the dingos). However, when a male dog mates with a coyote, unlike male coyotes the dog does not stay to help raise the pups, and the pups almost certainly die. This means that "dog genes" generally won't enter the coyote gene pool, and thus the two animals are a separate species. Dingos and dogs, however, can and will produce viable offspring, and so they are both subspecies of dogs (dingos having been descended from domestic dogs that became feral about 8000 years ago). This does not mean you can pet the dingos.
Speciation with respect to creationism
Speciation - whether a species evolve enough to diverge into a new and different species - is one of the key points that anti-evolutionists reject. In this case, they lump the ability for speciation to occur into so-called macroevolution. This is because the evidence for "microevolution" is so mind-numbingly obvious that they really cannot ignore it so must shoehorn it into a revised model; one can say that it's the only way of resolving the cognitive dissonance produced by rejecting evolution while still seeing the most blindingly obvious evidence that we can live long enough to observe. The primary problem with rejecting the idea of speciation is that it is based around the idea that the defining boundaries between species are real, solid and not flexible - something that is certainly not observed in nature or advocated by modern biology (as described above). Because of the malleability of the term, and the fact that evolution causes gradual changes, we will never see speciation (as creationists like to think of it, at least) happen. Only in hindsight, after many changes have taken place and we remove the transitions, can we see that speciation has occurred.
Slightly - and we mean slightly - more on-the-ball creationists or intelligent design advocates have come up with the concept of baraminology, which on the face of it, is similar to higher levels in the conventional biological classification, such as genus or family. A "baramin" is therefore a group of animals that contains multiple species. Baraminology was originally formulated to try and solve the problem of the high number of animals that would be on Noah's Ark, but can also be used as an excuse to accept speciation when it is conclusively observed, while still rejecting evolution. However, a baramin, unlike the biological classifications, has very little in the way of an objective description and is usually classed by fairly simple terms such as "does it look horsey or fishy" - and more often than not is extremely flexible. Hence creationists can now accept speciation, so long as you don't cross into a different baramin, which are so broad that it would be impossible to view such a change in less than a few hundred thousand years at least.