Rabbi Eliyahu's great abilities began to show at a very early age. At the age of seven he gave his first public discourse and displayed a fully developed intellect. By the time he was ten he had advanced to the point where he no longer needed a teacher. When he was still a young man, Rabbi Eliyahu accepted upon himself "galus," self-imposed exile (a not unheard of practice at that time), in which he wandered from community to community as a beggar. This lasted for a period of some years whereupon he returned to the city of Vilna. Despite efforts on his part to hide his great righteousness and phenomenal knowledge, he was soon famed as a great tzadik (righteous man) and Torah scholar. At the age of 35 he was approached by one of the leading sages of that time, Rabbi Yonason Eybschutz, to act as an intermediary in the conflict between him and another great sage, Rabbi Yakov Emden.
To the Gaon, limud haTorah (the study of Torah) was of paramount importance and this ethic was fully displayed on his part. His diligence in learning was unsurpassable. The Gaon's son testified that for fifty years his father did not sleep for more than two hours in a twenty-four hour period. His breadth of knowledge was amazing. He was capable of stating from memory the number of times any sage was mentioned in any particular book of the Talmud. His knowledge of both the revealed and the hidden parts of the Torah was beyond compare. The Gaon considered secular knowledge to be a vital adjunct to Torah study. He was knowledgeable in almost all secular fields and authored books on grammar and mathematics.
His righteousness and kindness were also legendary. Despite his personal poverty he always gave 20% of his income to charity. When informed of a special need such as marrying off an orphan girl or redeeming a captive, he would frequently deduct money from his personal needs to contribute. Despite his constant diligence in learning he always kept an ear out for people in need and was known to interrupt his studies in order to meet with relatives of a person in need to convince them to help their relative out.
There is a story of the Gaon which illustrates the kind of kindness he was capable of. The city of Vilna paid a small monthly stipend to the Gaon. (It should be noted that the Gaon refused to accept any official position in the community despite the fact that he was generally viewed as the leader of the community.) The individual who was responsible for delivering this money would take this money for himself. The Gaon, who realized what was happening, never accused him of doing this nor told anyone of it because he did not want to shame the person responsible. Indeed, we would not even know of this incident if the guilty person had not confessed on his deathbed.
For forty years he studied in isolation, from that point on he began to take in students from the outstanding Torah scholars of that time. Most of the writings we have today from the Gaon were transcribed by these students. One of the most famous of the Gaon's students was Rabbi Chaim of Volozhin, founder of the Volozhiner Yeshiva. This yeshiva (Talmudical academy) was the premier center of Torah study for about 100 years and is the ancestor of most of the yeshivas that exist today.
Possibly the Gaon's single biggest contribution to the Jewish people was his corrective notes on most of our ancient texts, particularly the Talmud. Over the centuries errors had crept into the various texts due primarily to scribal mistakes. (We are talking here about ancient texts other than the Torah itself. The rules for writing the Torah are so strict that scribal error is nearly impossible and extraordinarily rare. Such mistakes as do occur rarely go unnoticed long enough to be duplicated into other scrolls.) These errors were serious obstacles to advanced study of the Talmud and other texts. The Gaon, with his phenomenal knowledge of the entirety of the Torah literature, was possibly the only individual capable of creating authoritative corrections of these texts. There is almost no ancient Torah text that does not bear the notes of the Gaon.
For many years the Gaon desired to travel to the land of Israel and settle there. The Gaon actually began the traveling at one point but was unable to complete the trip (the reason for this is unclear). It was during this trip that the Gaon wrote his famous letter back to his wife instructing her on various ethical issues such as educating their children during their separation. This letter has become a classic in it's own right. Ultimately, about ten years after the Gaon passed away, many of his leading students followed in their master's footsteps and settled in the land of Israel.
The Gaon was the leading opponent of Chassidus, a movement founded by Rabbi Yisrael Baal Shem Tov in the 1730s (approximately). The Chassidim instituted a number of changes in standard Jewish practice and many great leaders viewed them as a nascent breakaway sect from true Judaism in the manner of similar movements in the past. The emphasis of Chassidus on mysticism was particularly worrisome in this regard. Chassidus also placed a very strong emphasis on fervent worship. While this is well within the boundaries of traditional Judaism, many of the opponents of Chassidus, called "Misnagdim," misunderstood this emphasis as detracting from the importance of Torah study. (This error was not without basis; unfortunately, many Chassidim (followers of Chassidus) also fell into this error.) The reasons behind this great controversy are complex and it is really not possible to do justice to them in a short essay. To all intents and purposes the controversy ended in the early 1800s with the introduction of the anti-religious Haskalah movement which created a need for all religious Jews to form a common front.
Despite the Gaon's opposition to Chassidus he was widely recognized by all groups as the leading torah authority of his time. Indeed, when after the Gaon passed away certain individual Chasidim expressed happiness at the news of his death, Rabbi Shneur Zalman of Liadi, one of the most prominent leaders of Chassidus at that time, issued a public letter forbidding such statements and requiring his followers to speak of the greatness of the Gaon.
The Gaon passed away in 1797 leaving behind a tremendous legacy, both from his vast and varied writings on all Torah subjects and from his outstanding students who went on to spread Torah throughout the people of Israel.
Sources: Great Jewish Leaders