The definition of "species" is a tricky one. Depending on a person's focus and need for the definition, the idea of the species concept can be different. Most basic scientists agree that the common definition of the word "species" is a group of similar individuals that live together in an area and can interbreed to produce fertile offspring. However, this definition is not truly complete. It cannot be applied to a species that undergoes asexual reproduction since "interbreeding" does not happen in these types of species. Therefore, it is important we examine all of the species concepts to see which are usable and which have limitations.
The most universally accepted species concept is the idea of the biological species. This is the species concept from which the generally accepted definition of the term "species" comes. First proposed by Ernst Mayr, the biological species concept explicitly says, "Species are groups of actually or potentially interbreeding natural populations which are reproductively isolated from other such groups." This definition brings into play the idea of individuals of a single species being able to interbreed while staying reproductively isolated from each other.
Without reproductive isolation, speciation cannot occur. Populations need to be divided for many generations of offspring in order to diverge from the ancestral population and become new and independent species. If a population is not divided, either physically through some sort of barrier, or reproductively through behavior or other types of prezygotic or postzygotic isolation mechanisms, then the species will stay as one species and will not diverge and become its own distinct species. This isolation is central to the biological species concept.
Morphology is how an individual looks. It is their physical features and anatomical parts. When Carolus Linnaeus first came up with his binomial nomenclature taxonomy, all individuals were grouped by morphology. Therefore, the first concept of the term "species" was based on the morphology. The morphological species concept does not take into account what we now know about genetics and DNA and how it affects what an individual looks like. Linnaeus did not know about chromosomes and other microevolutionary differences that actually make some individuals that look similar a part of different species.
The morphological species concept definitely has its limitations. First, it does not distinguish between species that are actually produced by convergent evolution and are not really closely related. It also does not group individuals of the same species that would happen to be somewhat morphologically different like in color or size. It is much more accurate to use behavior and molecular evidence to determine what is the same species and what is not.
A lineage is similar to what would be thought of as a branch on a family tree. The phylogentic trees of groups of related species branch off in all directions where new lineages are created from speciation of a common ancestor. Some of these lineages thrive and live on and some become extinct and cease to exist over time. The lineage species concept becomes important to scientists who are studying the history of life on Earth and evolutionary time.
By examining the similarities and differences of different lineages that are related, scientists can determine most likely when the species diverged and evolved compared to when the common ancestor was around. This idea of lineage species can also be used to fit asexually reproducing species. Since biological species concept is dependent upon reproductive isolation of sexually reproducing species, it cannot necessarily be applied to a species that reproduces asexually. The lineage species concept does not have that restraint and therefore can be used to explain simpler species that do not need a partner to reproduce.