Courtesy was an act done as a kindness or formality.
The term common courtesy referred to basic standards of politeness that were expected in a society, typically taught a young age and therefore expected without explanation. Those new to a culture could struggle with such norms, as others might expect them to already know them, even when they were still learning.
In 2267, when Commander Spock noted that Khan Noonien Singh was making considerable use of the technical library of the USS Enterprise, Captain James T. Kirk described it as being common courtesy, given that Khan was displaced from the time in which he had originally lived and therefore it was decent of them to help him catch up. (TOS: "Space Seed")
In 2268, as Lokai said he would not return to Cheron with Bele, James T. Kirk said, "I told you where you were going. I allowed your fellow countryman here simply as a courtesy." (TOS: "Let That Be Your Last Battlefield")
Zaldans were infuriated by courtesy, viewing it as a form of phony social behavior that covered true feelings. Knowledge of this allowed Wesley Crusher to pass a secret test as part of the Starfleet Academy entrance exam. (TNG: "Coming of Age")
In 2371, Ensign Harry Kim told Lieutenant Tom Paris that the two of them should have invited Captain Kathryn Janeway to sit together with them in the mess hall of the USS Voyager. Paris, however, told Kim that ensigns did not invite captains to sit down, that it simply was not done, but Kim asked what was wrong with showing a little courtesy. Paris told him that captains did not want courtesy, they wanted respect and therefore did not get chummy with the lower ranks. (VOY: "The Cloud")
In 2375, when Ensign Tom Paris interrupted Lieutenant commander Tuvok while he was meditating, he asked him if there was a reason for his lack of courtesy. He replied that there certainly was, querying him on just what he had said to Noss, a woman who had an affection for him that he was unable to reciprocate as a Vulcan. When he was replied that their conversation was private, he told her that she was now talking about leaving and going off on her own, potentially placing her in danger. (VOY: "Gravity")