A symposium was an academic conference. The Annual Starfleet Symposium was an example of this.
Beverly Crusher met Jas Holza a few years prior to 2368 at a symposium. They danced together at a reception after the symposium. According to Ro Laren, Holza was invited to symposia and diplomatic soirées, but was without real influence among the Bajorans. (TNG: "Ensign Ro")
In 2366, the Astrophysics Center on Icor IX held a symposium on rogue star clusters, which Jean-Luc Picard considered to attend. (TNG: "Captain's Holiday")
The Federation Archaeology Council held an annual symposium. In 2367, this symposium was held aboard the USS Enterprise-D in orbit of Tagus III. (TNG: "Qpid")
During the Psychological conference in 2369, Geordi La Forge attended a warp energy symposium, where he was able to touch a plasma field. (TNG: "Timescape")
In 2377, the Ovions asked The Doctor to speak at a symposium about spaceborne pathogens. (VOY: "Flesh and Blood")
In 2378, The Doctor and Kathryn Janeway attended a medical symposium together. (VOY: "Renaissance Man")