Immortality, in general, is the state of living indefinitely. While precise details can vary, this may involve an inability to be killed and an indefinite life expectancy with regard to natural causes. The impact of this on mental health is sometimes speculated about.
In EGS, "immortals" (fairies) are entirely immortal and generally maintain sanity by voluntarily resetting periodically. They can also be forced to reset. This results in a complete regeneration, reminiscent to the immortal phoenixes of mythology, which does not inherently retain prior memories, although this can be worked around in the case of voluntary resets given due preparation. See that article for more information. They have generally been equated with the "ancients" of the griffins' side of the universe.
Half-immortals (elves) are (sterile) hybrids of immortals (fairies) and mortals. They have long lifespans due to their immortal parentage, it was not mentioned yet if indefinite. It was implied (but not directly confirmed) that they can be killed.
Aberrations are prolonging their lives by feeding on humans (or possibly other sapient beings). It is possible (but not confirmed yet) they can extend their lives indefinitely. They can regenerate even very serious woulds, however they can be killed by magic or magic weapon.