Abarim Publications' online Biblical Greek Dictionary
The noun αιων (aion) means life-span, age or epoch. It is the source of our English words age and eon, and ultimately stems from the Proto-Indo-European root aiw- meaning the same.
Our noun appears to describe any length of clearly defined span of time, whether one person's life or some era of certain renown. As general point of referral, this word may also apply to the vast span of history: the ages, the times of old. It's used 126 times in the New Testament; see full concordance
Note that the Greeks also applied this word to the spinal marrow, which appears to suggest that they saw society as much based on social evolution as a human being on his spinal cord. To the Greeks, time wasn't merely the stage upon which all things unfolded but realized that time is an inherent and fundamental quality of the universe that doesn't exist separate from it (something moderns didn't know until Einstein). They personified abounded but cyclical time, or "the ages" as the deity Aion and associated him with a coiling snake that often ate its own tail or else curled along a spoke or spine.
In Roman times, Aion began to personify the eternal Roman rule, which made him/it an obvious target of the New Testament writers (Matthew 6:13, Luke 1:33, Romans 1:25). The Roman Aion even begat a female counterpart named Aeternitas (from which we have our English word eternity) who adopted the patronage of the deified Roman Emperor. This in turn made her too an obvious topic of commentary for the earliest church fathers.
From our noun αιων (aion) comes the adjective αιωνιος (aionios), meaning eternal, or more precise: lasting an age. This word was also used to describe the term of an office when it was to be held for a life time (an αιων, aion). The familiar phrase "eternal life" uses our adjective in combination with the noun ζωη, zoe, which doesn't describe one's personal life but rather life as a phenomenon that arises from interaction of elements (precisely how time works). So no, the term "eternal life" does not describe one's personal perpetuation, but rather speaks of an epoch that consists of souls rather than seconds: the most perfectly intertwined social expression also known as the New Jerusalem (Revelation 21:2). Our adjective occurs 71 times; see full concordance.