From a Buddhist point of view, enlightenment (Bodhi - literally: "awakened") is a state of consciousness where one becomes aware of the truth of one's existence, which is tantamount since in Buddhism, one's existence is the truth of all existence. Hence the reason why when a brahmin questioned the Buddha about his seemingly unearthly appearance, asking if he was a god or a spirit, the Buddha replied no, that he was simply "Awake."
The term comes from Vedic philosophy and Hinduism (Sanatana Dharma) which precedes Buddhism and had a bit more to say on the subject. Whereas Buddhism considers the truth of one's existence to be tantamount, Hinduism holds that one's existence is a falsehood separated from the actual truth which is the Supreme Reality or Absolute Reality of Brahman (a concept that is removed from Buddhism).
In Hinduism, knowledge of Brahman (Brahmavidya) is the ultimate goal for it brings the gift Moksha, liberation from our false reality and reunion with the entity of Brahman. The Self is dissolved away and all that remains is Brahman. However, from a human perspective, complete knowledge of Brahman is an impossibility, and only after death is one ever truly capable of being reclaimed by the Absolute - or so the theory goes.
With this in mind, Bodhi or enlightenment in the original Dharmic tradition denotes a state of being where one has achieved the necessary knowledge to ascertain that there is indeed an Absolute Reality, even if one cannot see it completely as one would if they were a Jivanmukta (liberated soul). To put it into practical terms, if one was standing at a closed door and the Supreme Reality (Brahman) was on the other side, Brahmavidya would be knowing there was a door somewhere, being enlightened (Boddhisattva) would be seeing the door, a Jivanmukta would be someone who opened the door, and liberation (Moksha) would be walking through it.
This last definition of enlightened that is often borrowed by the West. A state of knowing that there is something beyond the readily observable.
Nirvana is the concept of absolute being which appears in the doctrines of the various Buddhist sects. It its the final goal for individual soul where you let go of all material attachments and enter into a state of pure consciousness, released from the conditional prison of samsara, the cycle of life, death, and rebirth. Achieving Nirvana involves the removing of your karmic burden, freedom from dukkha (dissatisfaction or suffering), and the realization (awakening / enlightenment) of the Absolute--the ground state of all reality. The Buddha taught that the path to Nirvana was the Middle Path between asceticism and hedonism.
Nirvana is a concept that is developed from the Hindu concept of Moksha (liberation from causality and differentiation--the false reality of dualism). In Hinduism, Nirvana has a slightly different meaning. It is not the final goal, but the step just before the final goal. Moksha involves union with the Absolute Reality that is Brahman and the dissolution of the Jiva (souls) and Atma (self). It means losing your identity and becoming a part of the foundation of the universe itself. In Hinduism, Nirvana means being aware of the absolute, but not necessarily being one with the absolute. Buddhism did away with this final step involving Brahman, making Nirvana the final step in their doctrine. It led to a huge shift in the politics of Hindu philosophy and religion at the time.