Immortality is a concept that a biological organism that never dies. Christ is considered to be a (God incarnate) person who is immortal.
A myth throughout the ages that there is a Fountain of Youth whose restorative waters could provide a means to immortality.
Some "immortalized cell lines", such as HeLa (for the person they came from, Henrietta Lacks, and her cervical cancer) are human cells that keep reproducing in the petri dish. Such cells are important in medical research.
There are promoters of the "life extension" or "anti-aging" movement such as Aubrey De Grey, who speculate that human life can perhaps become eternal.
Speculation on cultural immortality
Some of those who have advanced civilization achieve a sort of cultural immortality. For example, Napoleon commented that while his French empire might, with time, go the way of the Roman Empire, that he would be remembered for his system of laws and national administration via his Napoleonic Code. The same can be said for the Code of Hammurabi.
Scientists may seek a sort of immortality in their discovery and published papers. Astronomers, especially those who have a celestial object named after them, may be motivated in their work by such notions. The same for authors of self-published or professionally published books (in the modern era, those bearing an ISBN) and such may seek the same. Parents may seek a sort of immortality via their biological legacy, although this form is less remarkable.
One could argue that a quality biography or a remarkable obituary (such as those listed in The New York Times) help to provide a sort of immortality while most others have to settle for a grave marker and the memory of their descendant or students for as long as that lasts.