(covers information from several alternate timelines)
A wheel was a round device capable of rotating on its axis, facilitating movement or transportation, or performing labor in machines.
Wheels were essential to the drive of automobiles. As they were a key part of the movement of cars, trucks, vans and other automobiles. There were typically four wheels on the bottom of a vehicle, used for driving it, and a steering wheel used for changing direction. A tire lock was attached to wheels to prevent the theft of automobiles. (ENT: "Carpenter Street")
There were also games that made use of wheels, such as roulette. In the game, a wheel is spun on the game table, leaving it to chance as to where the wheel stops and the ball lands. (TNG: "The Royale")
In an alternate 2259, Admiral Christopher Pike attempted to explain to Captain James T. Kirk that the Nibirans were especially vulnerable to cultural contamination, as they had "barely invented the wheel." (Star Trek Into Darkness)
When Hikaru Sulu was telling the rest of the senior staff of the USS Enterprise about the USS Excelsior having transwarp drive, Montgomery Scott told him "if my grandma had wheels, she'd be a wagon." (Star Trek III: The Search for Spock)
In the 2370s, the Argo was a shuttlecraft attached to the USS Enterprise-E that was equipped with a buggy that was also called the Argo. It was held in place in the shuttle by clamps attached to the wheels. (Star Trek Nemesis)