(covers information from several alternate timelines)
A sphere is a round geometric shape that could roughly be described as a ball-shaped object.
The volume of a sphere was calculated as "four-thirds pi times the radius cubed." (Star Trek)
According to Odo, a sphere was "one of the most basic forms in nature," while demonstrating a marble to a sick infant Changeling. As he rolled the marble around the dish, he asked, "See how it rolls? Interesting, isn't it?" Odo then poured the Changling into a spherical glass and explained, "Now, this is also a sphere. Feel its symmetry, the softness of its shape, the sameness." (DS9: "The Begotten")
Spherical designs 
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The Borg created large spherical starships often used as a long range tactical vessel, known as Borg spheres. (Star Trek: First Contact) The twenty-two fragments of outer hull wreckage discovered under the ice in Earth's northern polar region was described by Rooney as all having "the same curvature", concluding that "I think this ship was a perfect sphere," that measured "[a]bout 600 meters in diameter." (ENT: "Regeneration")
There were a series of artificially-created spherical objects that were scattered around the Delphic Expanse in the mid-22nd century to distort space to create an environment that could support the Sphere-Builders. (ENT: "The Forgotten")
The Xindi created a massive spherical device in 2153 to be used against Earth. (ENT: "The Expanse")
A Dyson sphere was a colossal sphere constructed around a star, completely surrounding it. The USS Enterprise-D encountered one of these spheres in 2369. (TNG: "Relics")
In describing the Altonian brain teaser to Julian Bashir, Jadzia Dax explained that the puzzle responded to neural theta waves with the goal being to turn the sphere into a solid color. This was achieved by sitting comfortably releasing all tension, and focusing one's concentration on the sphere itself. (DS9: "A Man Alone")
That same year, Klingons discovered Saltah'na energy spheres, created to store the essence of a personality. (DS9: "Dramatis Personae")
Harry Kim explained to to an amnesiac Tuvok that the object of the kal-toh was "to turn this jumble of rods into a perfect sphere. We take turns positioning our pieces. Whoever gets the shape to appear first, wins." (VOY: "Riddles")
