Deoxyribonucleic acid (DNA) is the building block of life. It is the blueprint for a living organism and encodes amino acids for polypeptides which form proteins, which, when properly sequenced, form cellular pathways that together form the physical structure and anatomical characteristics of an organic lifeform.
If DNA is changed in some way, the host organism responds to the change by re-making itself in the form specified by the new DNA, often extremely rapidly. Semi-corporeal beings such as the Founders also had DNA though there is no evidence they had any cellular structure or indeed any cells.
In 2369, following clues developed by his mentor, Richard Galen, Jean-Luc Picard discovered evidence that an ancient race of humanoids had seeded many planets with their DNA billions of years earlier. The evidence suggested that Humans, Romulans, Klingons, Cardassians, and others may have far more in common than previously believed. (TNG: "The Chase")
DNA extracted from a male could be used to impregnate a female. In 2372, Seska claimed she impregnated herself with DNA she had stolen from Commander Chakotay while he was her prisoner. (VOY: "Maneuvers")
As DNA was relatively unknown in the 1960s, Star Trek: The Original Series has no mention of this rather significant piece of medical science, even though DNA would have been well known in the 2260s according to Star Trek canon. It is also for this reason that Original Series episodes depended on other medical methods, whereas DNA samples would have been the preferred choice. Such examples include the voice print identification of Kodos, seen in TOS: "The Conscience of the King", and a reference to "medical plates" during the examination of Peter Kirk in TOS: "Operation -- Annihilate!".