Creationists make much of the fact that biologists have been unable to satisfactorily define what a species is. Since, after all, the concept of a species is part of the 'evolutionist lingo', isn't it? I agree, it has become embedded in the lexicon of biology in general and when evolution is the subject of discussion, the term 'species' is likely to be used. The history of the term 'species' in biology may surprise you.
The term 'species' is actually a creationist concept.
It was formalized by the father of taxonomy, Carl von Linné (Carolus Linnaeus) an 18th C. Swedish naturalist. The concept of a species was meant to embody the original created kinds (I've used the term kind here on purpose which will become evident later). The representatives of species today were supposed to be descendents of the original pairs created by God on creation week. The original pairs were the original Platonic archetype that formed the essence of the species. This has led to a long-established tradition that continues in taxonomy and museum collections to this day: the type specimen. Type specimens are meant to represent the platonic archetype of the species and form the reference for all other studies on the taxonomy of that species. Species were unchanging, except in that they were only an approximation of their original ancestor.
Just as we could not build a chair that was the archetypal chair (it would always be in some way imperfect), we could not have individuals that were the archetypal species. However, they never deviated far from their essential archetype.
Darwin changed all that.
He showed that a species was not as clear cut as we might have thought. As taxonomists were faced with difficulties (and often contradictions) in their classifications of 'species', Darwin sought an explanation. The first thing he did was point out that species were difficult to classify because the boundaries which we assumed were there were, in fact, not there. Species often graded into one another, or were separated by trivial characters that were variable from one individual to the next. In essence, species as we knew them, were not real.
Evolutionary biology aims to explain how the diversity of life arises. The reality is that evolutionary biology came to be when a system of classification of that diversity was already established - based on creationist concepts. This diversity is classified into 'buckets' of variation. The smallest buckets (and therefore should include the most limited variation) is the species. Traditional taxonomy places species within genera, genera within families etc. Ergo, the task of evolutionary biology is to explain the origins of these buckets - the origin of species, genera, families, classes...
The historical reality is that evolutionary biology has inherited a creationist concept and tradition of taxonomy. However, by Darwin's time, that tradition had become so well established that nobody thought to reorganize it. Museums all over the world were loaded with hundreds of thousands (if not millions) of type specimens and counting. The task of overturning this infrastructure would have been colossal. I doubt it would have crossed anyone's mind!
So today, evolutionary biology deals with terms that are conceptually different from the definitions of their historical antecedents. 'Species' doesn't mean what it meant 200 years ago. The reality that biologists have to deal with is that no single, universally satisfactory definition of a species seems to exist. This has been the problem ever since Darwin.
The fact that creationists point out that biologists have been unable to define a species is ironic. If evolution is true, and species are related to one another, grades between species and the difficulties in drawing boundaries between groups ought to be observable. We can't have infinitely many gradations, as some creationists request, because the world cannot support infinitely many species any more than it can support infinitely many individuals. The problem of such definitions is one of the principle reasons why Darwin thought that evolution was in fact occurring.
Why do creationists proffer this challenge, then? Well, it seems to stem from a psychology of knee-jerk antagonism. It doesn't seem to matter what, as long as evolutionists face a challenge or a problem, creationists wish to point it out and spin it as though this were somehow evidence of a theory in crisis. It is, in fact, evidence of a theory at work.
The challenge usually comes out when evidence of species transformations in real-time are provided. Their reaction to this is that this is merely variation 'within a kind'. God only said he created the animals after their 'kind', but they were free to change somewhat after that. For instance, two species of island finch evolving is not the same as 'a fish turning into an amphibian' (as though evolution proposes that happened all at once). So, we generally ask them to define what a 'kind' is. Instead of responding in a meaningful way, creationists generally avoid answering by saying 'you can't define what a species is, either!'
No kidding. It was your lot that foisted this notion on us and we have dealt with it appropriately, showing that it doesn't exist and that species do in fact change. Now, the goalposts have slid one level of the hierarchy higher (conveniently to a level beyond a normal 'ecological time span' - or within the lifetime of a human being). The 'kind' is just a species. Whenever real-time evolution is pointed out, that will always be evolution 'within the same kind'. Creationists are here resorting to a modified version of archetypes, which has been shown repeatedly by ecology, genetics, and the fossil record not to exist in biology.
In sum, the concept of a species - indeed all of traditional taxonomy - is rooted in creationist and platonic concepts of static archetypes. Species definitions are difficult to obtain because species are not really bounded by these kinds of typological boundaries. Species, as we recognize them, are in historical continuity with each other and are therefore difficult to tease apart, especially when their divergence was relatively recent or ongoing. So, we have creationists to thank for all the problems that come with the concept of 'a species'. It's not the problem of but part of the cause of evolutionary biology.