Relatively little is factually known about Elisha ben Abuyah (alternate spellings abound, commonly "Avuyah" or "Avuya"; it's a transliteration of ancient Hebrew, which remained an unspoken language for centuries, so it's inexact at best). Most of what we know comes from Mishnaic and Talmudic stories and legends, which are often intended as allegory or to teach a point, not to be accurate.
We do know he was born around the time of the rule of the Roman Emperor Vespasian (circa 70 CE, the time of the destruction of the First Temple), in the Galilee in Israel to one Elishebah, the wife of Abuyah ("ben Abuyah" means "son of Abuyah"); his mother died shortly thereafter.
It is not easy to decide as to his exact attitude towards Judaism. That he came to refuse to accept the current rabbinical views is certain, though the Talmud cites his legal decisions. It is apparent that although he was at first considered a great Rabbi and elder, he did something that was considered a heresy against the leadership of first-century Palestinian Jewry. Most authorities believe that he became a Gnostic, although there are other Talmudic legends that support the claim that he became an atheist and it has been plausibly claimed that he became a Sadducee (the Sadducees were a sect of early Jews that denied the validity of any religious doctrine but the Old Testament and did not accept Rabbinical rulings). The claim put forward by some that he converted to Christianity is largely unfounded, and has gained very little support in recent years.
His disciple, the famous Rabbi Meir, remained his steadfast friend despite Elisha's later heresy; the Talmud gives accounts of a few of Meir's attempts to get Elisha to repent, to no avail. Whatever precisely he did, it was bad enough that the Talmud later refuses to mention him by name and refers to him only as "the Other" from then on. He came to be regarded as the type of a heretic whose pride of intellect betrayed him into infidelity to law and morals. Without much appropriateness Elisha has been sometimes described as the "Faust of the Talmud." His time and circumstances of death are uncertain; it was likely sometime in the middle 2nd century. His status as an "intellectual outlaw" made him a historical hero to the Haskalah ("Jewish enlightenment") movement of the 1800s.
A beautiful, depressing, moving, and somewhat long account of his life is given in As A Driven Leaf by Milton Steinberg; it is mostly fictionalized, however. Well worth a read if he seems like an interesting character to you.
The following are portions of Jewish texts having to do with Elisha ben Abuyah; realize they are not all completely accurate (some are nearly unintelligible); the Talmud abounds, for example, with stories of spirits coming from the afterlife to talk to colleagues, mystical recipes for a device to see spirits using the ashes of a black cat, and the like. Naturally this list is far from complete; if anyone knows any references I've missed, please let me know. (added note (3/4/03): while most of the following references to 'Acher' ('the other') are generally held to be talking about Elisha, the term is used in the Talmud in other places as well, so some of these stories may not have originally been about Elisha (although most have become part of the legend of the man nowadays regardless)).
Elisha ben Abuyah said: He who learns as a child, what is he like? Like ink written on new paper. And he who learns as an old man, what is he like? Like ink written on blotted paper. (Ethics of our Fathers, 4:20)
Four men entered the Garden (usually interpreted as the Kabbalistic realm of theosophy and hidden teachings). Ben Azzai died; Ben Zoma went mad; Acher (Elisha) destroyed the plants (an idiom for "commited heresy"); R. Akiva departed unhurt. (Babylonian Talmud Hagigah 14b)
The Other (Elisha Ben Abuyah) cut down his own offspring. Of him scripture says: "Do not let your mouth make your flesh sin" (Ecclesiastes 5:5). What is this about?
He saw that Metatron happened to be granted authority to sit to record the merits of Israel, and he said: "It is a tradition that on high there is no sitting and no strife,
no division and no toil. (So) perhaps there are two supreme Powers." Heaven forbid! Then they brought him to Metatron and they smote him with sixty bands of fire.
They said to him: "When you saw him (Metatron), why did you not stand up before him?" Then authority was granted (Metatron) to erase the merits of the apostate. A heavenly voice went forth and said: "'Return you turncoat sons!' (Jer 3:22), except for this apostate." (Babylonian Talmud Hagigah 15a)
"He (Elisha ben Abuyah) found a whore, and wanted her. But she said, "Aren't you Elisha?" So he violated Shabbat by picking a radish from the ground and gave it to her, and she said, "He is Acher ('another one')." (Jerusalem Talmud, Hagigah 2:1. Note the pun - who said first-century Rabbis didn't have a sense of humor?)
When Elisha ben Abuyah was about to die, Rabbi Meir came to visit him and urged upon him to repent. Asked Elisha: "And if I repent, will I be accepted?" Answered Rabbi Meir: "It is written (Ps. 90:3): 'You turn man back to contrition and say: 'Return you sons of men.'' Up to the very last flicker of life repentance is accepted." Thereupon, Elisha wept and his soul departed. And Rabbi Meir rejoiced in his heart, saying: "It seems that my master passed away in the act of repentance." (Jerusalem Talmud Hag 2:1)
One of the most depressing Talmudic stories relates that he once saw a child climbing a tree to shoo away a mother bird so he could take the eggs upon his father's order; sending away the mother bird and honor thy father and mother are the two mitzvot with the reward listed in the Bible as "long life." The child fell to his death, and he concluded "there is no justice and there is no Judge." (Talmud Kiddushin 39b).
Other assorted Talmudic sources relate similar stories of his losing faith upon seeing the tongue of a great Rabbi being dragged by a dog or a pig; others say that he kept heretical books and was worldly and sang Greek songs.
the Jerusalem and Babylonian Talmud
Encyclopedia Judaica, 1972 edition
"As A Driven Leaf" by Milton Steinberg
1911 Encyclopedia Britannica