The biological species concept is limited to describing sexually reproducing populations that are currently alive. Due to the precise language used to articulate the concept, and to define "species," it is unsuited to understanding asexual or extinct organisms.
The modern biological species concept was advanced by Ernst Mayr in 1942. Mayr defined a species as any group of organisms that currently or potentially reproduce with each other. This is best understood as any group of living things that could combine their genes if they are both healthy, sexually mature and in proximity to each other. This concept provides a useful tool for understanding most animal and many plant species.
The concept breaks down, however, for those organisms which don't need partners to reproduce. While two such organisms might be morphologically very similar, their inability to breed technically makes them members of distinct species. The reverse is true of ring species, in which a geographical barrier partially separates a population in such a way that genes flow freely all along the chain, but individuals at either end are too different to breed with each other directly. Prokaryotic microbes also defy species classification, as they often share genetic material across what would otherwise be species barriers. Finally, extinct organisms cannot be truly grouped into species, as they are no longer able to breed, which creates controversy in the classification of fossils.