(covers information from several alternate timelines)
The singing plant was a plant native to Talos IV. It was a blue plant with a single, broad lily pad-shaped leaf on a long stem, and sometimes featured a white flower on top. It could be found growing among the many boulders and plains that made up the planet's surface.
As its structure vibrated in the near constant winds on the planet's surface, the Talosian singing plant emitted a beautiful, wailing chime, which was very pleasing to the ear. Holding the plant's leaf still in one's hands would cause the song to cease, and cause the plant to close up, but it would resume again once let go.
Terran emperor Philippa Georgiou told Captain Leland that in the Terran universe, she destroyed the Talosians and their "stupid singing plants". (TOS: "The Cage", "The Menagerie, Part I"; DIS: "If Memory Serves")
The script of "The Cage" describes the noise produced by this kind of plant as "a musical flute-like sound" and characterizes the plant itself as "a plant with paper-thin rigid leaves which vibrate in the breeze. [Spock] blows on it, produces new musical notes." The script continues with a bracketed production note that reads, "These wind-plant sounds will identify and set mood for all our scenes on the surface of Talos IV."