In geometry, two-dimensional space was a projection of the physical universe onto a plane. The two dimensions of this space were length and width.
In 2259, after experiencing the intoxicating effects of Orion hurricanes, the crew of the USS Enterprise hallucinated that they were two-dimensional cartoons with large eyes and elastic arms. (SNW: "Those Old Scientists")
During the Battle of the Mutara Nebula in 2285, Spock noted how Khan Noonien Singh patterns indicated "two-dimensional thinking" in three-dimensional space. To use this to their advantage, Admiral James T. Kirk lowered the altitude of the USS Enterprise by 10,000 meters. (Star Trek II: The Wrath of Khan)
In 3189, Commander Michael Burnham suggested, with three points of data, that she and Ensign Sylvia Tilly could triangulate and pinpoint the source of The Burn. Tilly responded to the suggestion by saying that it might work in a two-dimensional space; however, it would not work in a three-dimensional space until they had more data. (DIS: "Unification III")