The carotid arteries were responsible for delivering blood containing oxygen and nutrients to the brain in most humanoid species. There was one carotid artery on each side of the typical humanoid neck. This artery was often used as the injection point for hyposprays containing intra-arterial medication. The size of these arteries, and their proximity to both the surface of the skin (at the neck) and to the brain made them an excellent choice for most such medications.
When Khan Noonien Singh was revived from suspended animation in 2267, he held a knife to Leonard McCoy's throat and demanded to know where he was. McCoy refused to answer, but offered a suggestion to Khan, that it would be most effective to slice the carotid just under his ear. (TOS: "Space Seed")
While Chakotay was hallucinating when the USS Voyager was trapped in chaotic space in 2375, a vision of The Doctor told him that a hard blow to the neck from an opponent in a boxing match would sever the carotid artery and cut off blood flow to the brain. (VOY: "The Fight")