There are some really good responses here. I'd like to offer this text by Thanissaro Bhikkhu, a comprehensive exploration of the concept of nibbana.
Mind Like Fire Unboundhttp://www.accesstoinsight.org/lib/auth ... index.html
Even just reading the abstract greatly helped me understand this concept, and diving into the more involved parts gave me a deep appreciation for practice towards liberation.
Here are some key excerpts from the abstract:
"Any discussion of the way the Buddha used the term nibbāna must begin with the distinction that there are two levels of nibbāna (or, to use the original terminology, two nibbāna properties). The first is the nibbāna experienced by a person who has attained the goal and is still alive. This is described metaphorically as the extinguishing of passion, aversion, & delusion. The second is the nibbāna after death. The simile for these two states is the distinction between a fire that has gone out but whose embers are still warm, and one so totally out that its embers are cold. The Buddha used the views of fire current in his day in somewhat different ways when discussing these two levels of nibbāna, and so we must consider them separately.
"To understand the implications of nibbāna in the present life, it is necessary to know something of the way in which fire is described in the Pali Canon. There, fire is said to be caused by the excitation or agitation of the heat property. To continue burning, it must have sustenance(upādāna). Its relationship to its sustenance is one of clinging, dependence, & entrapment. When it goes out, the heat property is no longer agitated, and the fire is said to be freed. Thus the metaphor of nibbāna in this case would have implications of calming together with release from dependencies, attachments, & bondage. This in turn suggests that of all the attempts to describe the etymology of the word nibbāna, the closest is the one Buddhaghosa proposed in The Path of Purification: Un- (nir) + binding (vāna): Unbinding.
"To understand further what is meant by the unbinding of the mind, it is also important to know that the word upādāna — the sustenance for the fire — also means clinging, and that according to the Buddha the mind has four forms of clinging that keep it in bondage: clinging to sensuality, to views, to precepts & practices, and to doctrines of the self. In each case, the clinging is the passion & desire the mind feels for these things. To overcome this clinging, then, the mind must see not only the drawbacks of these four objects of clinging, but, more importantly, the drawbacks of the act of passion & desire itself."
"The mind at this point attains Deathlessness, although there is no sense of 'I' in the attainment. There is simply the realization, 'There is this.' From this point onward the mind experiences mental & physical phenomena with a sense of being dissociated from them. One simile for this state is that of a hide removed from the carcass of a cow: Even if the hide is then placed back on the cow, one cannot say that it is attached as before, because the connective tissues that once held the hide to the carcass — in other words, passion & desire — have all been cut (by the knife of discernment). The person who has attained the goal — called a Tathāgata in some contexts, an arahant in others — thus lives out the remainder of his/her life in the world, but independent of it.
"Death as experienced by a Tathāgata is described simply as, 'All this, no longer being relished, grows cold right here.' All attempts to describe the experience of nibbāna or the state of the Tathāgata after death — as existing, not existing, both, or neither — are refuted by the Buddha. To explain his point, he again makes use of the metaphor of the extinguished fire, although here he draws on the Vedic view of latent fire as modified by Buddhist notions of what does and does not lie within the realm of valid description."
"The Buddha borrows two points from the Vedic notion of fire to illustrate this point. Even if one wants to assume that fire still exists after being extinguished, it is (1) so subtle that it cannot be perceived, and (2) so diffuse that it cannot be said to go to any one place or in any particular direction. Just as notions of going east, west, north, or south do not apply to an extinguished fire, notions of existing and so forth do not apply to the Tathāgata after death.
"As for the question of how nibbāna is experienced after death, the Buddha says that there is no limit in that experience by which it could be described. The word 'limit' here is the important one. In one of the ancient Vedic myths of creation, the universe starts when a limit appears that separates male from female, sky from earth. Thus the implication of the Buddha's statement is that the experience of nibbāna is so free from even the most basic notions making up the universe that it lies beyond description. This implication is borne out by other passages stating that there is nothing in that experience of the known universe — earth, water, wind, fire, sun, moon, darkness, coming, going, or stasis — at all.
"Thus, when viewed in light of the way the Pali Canon describes the workings of fire and uses fire imagery to describe the workings of the mind, it is clear that the word nibbāna is primarily meant to convey notions of freedom: freedom in the present life from agitation, dependency, & clinging; and freedom after death from even the most basic concepts or limitations — such as existence, non-existence, both, or neither — that make up the describable universe."