General Conference is literally the oldest tradition of the LDS Church, as the church was formally organized at a General Conference meeting on April 6, 1830. Section 20 of the Doctrine and Covenants lays out the basic organizational details for the church established during that meeting. Of particular interest to this question are verses 61 and 62:
61 The several elders composing this church of Christ are to meet in
conference once in three months, or from time to time as said
conferences shall direct or appoint;
62 And said conferences are to do whatever church business is
necessary to be done at the time.
The practice of quarterly conferences was changed fairly early on, as the rapid growth of the Church made it impractical. Traditionally conference lasted for three days, and the April conference would always include the anniversary of April 6, which often proved difficult when the scheduling conflicted with weekday commitments to work or school. This was changed in April 1977 and today they are held on a semi-annual basis, the first weekend of April and of October.
The general public is openly invited to General Conference. It is streamed live on the Internet, in addition to being broadcast by television, radio, cable and satellite throughout the world. All stake centers (meetinghouses designated as the administrative center of a stake, or collection of local congregations) and many ordinary meetinghouses have satellite dishes or other equipment installed to receive these broadcasts, and display them to the Church members or any members of the public who wish to stop by.
The format of General Conference tends to be fixed, consisting mainly of sermons delivered over the course of six two-hour sessions by top Church authorities. Church business is also conducted, such as the calling or release of leaders and a statistical reports on the state of the Church "by the numbers". As is common in LDS meetings in general, hymns are sung at every meeting, by various choirs from the Church. It is common to have the Mormon Tabernacle Choir provide the music for at least one session. However, all meetings fall under the guidance of verse 45 of the aforementioned charter in Doctrine and Covenants section 20, which states that those conducting meetings are to do so "as they are led by the Holy Ghost," meaning that changes can be made as necessary. This was done quite memorably on at least one occasion. (See below.)
Notable moments in General Conference:
April, 1830: The organization of the Church of Jesus Christ of Latter-Day Saints.
October, 1856: Brigham Young, upon receiving word just before the start of Conference that a large group of Mormon emigrants was stranded by a snowstorm with inadequate supplies and people were dying, essentially called off the whole conference. He claimed that the Holy Ghost had "dictated" to him that the highest priority at the moment was "to save the people," that to live by their beliefs was far more important than to talk about them. He gave one short sermon in which he explained the issue and ordered the organization of a rescue party to be made ready and depart immediately, and then dismissed the conference so the members could get to it. (The rescue party headed out later that day and was largely successful.)
October 1890: President Wilford Woodruff read he called "the Manifesto," an official declaration putting to an end the practice of plural marriage in the Church.
April 1998: President Gordon B. Hinckley announced a new plan for building temples at a much smaller scale. Previously, temples, considered the House of the Lord, had been rather palatial in size and scope, (with the best-known example being the iconic Salt Lake Temple,) and correspondingly expensive to build and maintain. Less than fifty had been constructed by the Church since its founding in 1830. By setting up a new option to build more minimalist temples, plans were made over the course of the next year to bring the total up to 100 or more by the end of the 20th century, bringing the blessings of temple worship to many more people throughout the world. By the time of President Hinckley's death in 2008, there were 124 temples in operation and several more under construction, due in large part to this "small temples" program.
October 1998: President Hinckley, gave a strong warning about debt from the pulpit. He specifically mentioned the importance of buying affordable homes, and he quoted from the Genesis account of Pharaoh's dream of seven years of prosperity followed by seven years of famine, ending on the warning from verse 32: "And God will shortly bring it to pass." He stated that he was "not predicting years of famine in the future," but nevertheless it was very important for Church members to "get their house in order" and get out of debt.
October 2001: Three years later, President Hinckley reiterated this warning, again invoking Pharaoh's dream, (and this time, notably, not saying that it would "shortly" come to pass,) and urged members to pay down their debts and specifically to pay down mortgages quickly. At the end of September of 2008, almost exactly seven years after this talk was given, the housing market imploded, bringing an end to a largely prosperous time and plunging the entire world into a severe economic crisis that, political optimism notwithstanding, it has yet to recover from. Church members who heeded their prophet's warning and made the sacrifices necessary to pay their mortgages off (or at least down) quickly were affected far less than most people.
October 2012: President Thomas S. Monson announces a change in the Church's missionary service policy:
I am pleased to announce that, effective immediately, all worthy and
able young men who graduated from high school... regardless of where
they live, will have the opportunity to be recommended for missionary
service... [And] able, worthy young women who have the desire to
serve may be recommended for missionary service beginning at age 19
instead of age 21.
He stated that moving the potential age of missionary service back allowed for greater flexibility, to make it easier to allow members to serve a mission without interfering with other obligations, such as national military service requirements.