A razor was a device used to remove body hair, most often associated with shaving beards. (ENT: "Unexpected") Types of razors include the manual straight razor and the electric shaver.
Lieutenant Malcolm Reed was of the opinion that an officer at his best should always be well-groomed. When he and Commander Charles Tucker III were stranded in Enterprise NX-01's Shuttlepod 1 with no hope of being rescued in time, Reed still insisted on shaving his beard with a razor so that his body would look well-groomed when found. (ENT: "Shuttlepod One")
In 2153, Henry used a straight razor to shave Sheriff MacReady. (ENT: "North Star")
In early 2364, Lieutenant Commander Data gave Lieutenant Geordi La Forge a razor that he had adjusted to perfect efficiency. La Forge, however, preferred his regular razor, as he was of the opinion that shaving was a Human art form and that technological perfection could shave too close. (TNG: "Code of Honor")
La Forge's razor is described in the script as "a glowing two-inch blue cube in his hand which he moves up, down and across his face an inch or so away from the skin". [1] Rick Sternbach designed this razor. (Star Trek: The Magazine Volume 2, Issue 12, p. 45)
Captain Benjamin Sisko used a razor similar to La Forge's to shave his bald head from time to time. (DS9: "The Sword of Kahless")
Lieutenant Commander Tuvok used a straight razor adorned with alien writings. In an alternate timeline of 2374, he was permanently blinded, making shaving a difficult task. After cutting himself on more than one occasion, Seven of Nine offered to help him, but he refused. (VOY: "Year of Hell")
In 2375, Counselor Deanna Troi shaved Commander William T. Riker's beard with a razor blade, while the two of them bathed in a bath tub on the USS Enterprise-E. He later joked to Lieutenant Commander Data that it made his face feel "as smooth as an android's bottom", though Data disagreed. (Star Trek: Insurrection)