As a rule, the word "immortal" is rarely used literally. Indeed, it can't be literally defined; you can never truly say that someone will never die, just that they haven't died yet. So generally it's used to mean a being that doesn't undergo aging and natural death the way most beings do. But it's usually assumed that such immortal beings can be killed by at least certain acts of violence. The classic example is vampires; they're theoretically able to live forever, but only if they avoid stakes to the heart, decapitation, sunlight, garlic, holy water, or a bunch of other things they're vulnerable to. (Although the sunlight thing was coined by the silent movie Nosferatu; vampires before that, and some since, have been unharmed by sunlight.)
Although there are various classes of immortal character. Some just live indefinitely unless they fall prey to deadly violence, like Tolkien elves. Others have regenerative ability that lets them heal completely from otherwise mortal wounds, like Flint, Vandal Savage, the Immortals of Highlander, or Wolverine (although it's unclear whether he's truly immortal or just slow-aging; he does have gray hair in "Days of Future Past"). But generally it's assumed that an extreme enough injury would be terminal, such as in the case of decapitation or being blown up for a Highlander immortal. Then there are the extreme cases like Captain Jack Harkness of Doctor Who/Torchwood where absolutely nothing can kill them; Jack even comes back from being blown to pieces. Or Demona and Macbeth from Gargoyles, who are cursed so that neither can die from any cause unless one of them kills the other.
In short, some "immortals" are more immortal than others, so the answer to the title question is "It depends."