Immortality

"Believe me, Captain, immortality consists largely of boredom."

- Zefram Cochrane in the Star Trek episode "Metamorphosis".

An Immortal is an entity that does not die by natural means, and in some legends can never die, even if disembodied.

Overview
In real life, biologically-immortal species are those which simply never die of old age, but are still susceptible to death from injury, disease, accidents, predation, poisoning and so on. In science fiction and fantasy, some creatures may achieve what basically amounts to immortality due to extremely advanced regeneration and/or technology; whereas others are truly immortal in the more supernatural sense of being impossible to kill (for instance, eldritch beings who will simply manifest onto a new physical body if their current one is destroyed).

Examples

 * The First Born from Babylon 5 are naturally immortal, though since they evolved over 7 billion years ago, most have perished from non-natural causes by now.
 * Being inorganic life forms, the Black Clouds from Fred Hoyle's The Black Cloud do not age naturally, and can potentially live forever. Their only method of reproduction is to transfer parts of their neural structure to regular nebulae, thus transforming them into others of their kind.
 * Being a composite organism made up of millions of microscopic components, the Giant Polyp from Arthur C. Clarke's The City and the Stars is essentially immortal: while individual components may perish, its gestalt conscience and memories are kept alive through the ages.
 * Bazin from Larry Niven's "The Death Addict" has achieved biological immortality thanks to nanosurgery. Although his potential lifespan is unlimited, he can still be killed.
 * The Immortals from Highlander are actually not immortal on their home planet, but naturally become so on Earth.
 * Wowbagger the Infinitely Prolonged from Douglas Adams' Life, the Universe and Everything is an alien who, although he wasn't born an immortal, had "his immortality thrust upon him by an unfortunate accident with an irrational particle accelerator, a liquid lunch and a pair of rubber bands". Unable to cope with eternity, he decided to find a purpose in life by insulting every other being in the universe in alphabetical order.
 * The strange, silicon-based Martian creatures known as Pyramid-builders from Stanley Weinbaum's "A Martian Odyssey" are biologically-immortal and, while they still reproduce, their seeds take millennia to develop. They are, however, practically mindless creatures which spend their lives away consuming silicate and building pyramid-shaped structures.
 * The Sintillians from Men in Black are described as immortal due to the high effectiveness of their two hearts.
 * The non-humanoid species known by the physiological classification EPLH from James White's Sector General are naturally long-lived by most species' standards, but their medical technology has advanced to the point that they've become effectively immortal, casually partaking in sociological projects that span for centuries or even millennia. Curiously, the rejuvenation technique they use also results in loss of long-term memory, so while they live for ages, they cannot recall anything from their early lives.
 * The Deep Ones from H. P. Lovecraft's "The Shadow Over Inssmouth" are a classic example of unaging, biologically-immortal beings.
 * The Q from Star Trek are so completely immortal that they cannot die even if they want to. The only way for a Q to die is to be transformed into a mortal by other members of his species.
 * Angels, spirits and demons are typically thought as immortal in most stories. Immortality is also one of the most common features of the gods, even though this isn't universal. In some systems of belief, the gods can still be killed, such as in Norse mythology.
 * Being undead beings, Vampires are generally regarded as unaging and thus biologically-immortal, although they can still be destroyed by several means (impalement and decapitation being the most well-known).
 * Examples of immortal Humans notably include: Koschei the Deathless from Russian folklore, Captain Jack Harkness from Doctor Who, Nicolas Flamel from Harry Potter, Orlando from The League of Extraordinary Gentlemen, John Oldman from The Man from Earth, Flint from Star Trek and Alfred Fellig from The X-Files.