Immortality in Tolkien's work is the concept of living for an infinite length of time, unless suffering a violent death.
Only beings such as the Ainur (including Wizards) and Elves are immortal. Men, Dwarves, Hobbits and most other races cannot live forever and thus are subject to aging and natural death. In Tolkien's universe, despite many having a desire for it, immortality is not desirable for those who are of mortal kind. Every race has a set span; to exceed this span proves to be agony for mortals. There are many examples of this throughout his work: the kings of Numenor who became witless and agonized by attempting to cling to life as long as possible (as opposed to earlier kings who gave up their lives willingly); Gollum's transformation into a twisted and starved thing after having his life extended nearly 5 times longer than it should have been via the One Ring; Bilbo's comment on his 111th birthday about feeling "thin and stretched", like "butter being spread on too much bread", this also being an effect of the One Ring. The only time mortals have ever been accorded peace in immortality is when they are accounted of the Elves: this choice has been mostly restricted to the Peredhil, the Half-Elven (the one known exception to this being Tuor, who was born a full-blooded Man but was given the choice to be accounted among the Eldar).
In the latter half of the Second Age, Sauron's corruption of the Numenorian King Ar-Pharazôn made him believe that if he could invade and conquer the Blessed Land he could become immortal, resulting in the destruction of Numenor and Ar-Pharazôn's imprisonment.