Spock (Leonard Nimoy) holds the rank of Commander based on the rank stripes on his uniform (two solid gold bands), however there are several times throughout the series where he is referred to as Lieutenant Commander. However a Lieutenant Commander's rank stripes consist of one solid gold band and one dashed/segmented band.
The Star Trek Universe is inconsistent in its use of measurement standards. In Star Trek: Obsession (1967), Ensign Garrovick describes a gaseous creature as being "several cubic meters" in size, but his distance as "about twenty feet away". These discrepancies continue through all the Star Trek series and movies.
Pavel Chekov's name should appropriately be spelled Chekhov, to accurately transliterate this common Russian name (see Anton Chekhov). The "kh" is a transcription of the Russian X, which makes a sound similar to but distinct from the Russian K.
It takes a while for character names and back histories to settle during the first season and there are many inconsistencies. At the same time, the crew wears different colors of uniform for the first few weeks.
In several episodes, the positions of people being beamed up/down change. For example, in Star Trek: Mirror, Mirror (1967) when the landing party is being beamed up, Kirk's arm is positioned down, then up as he starts to materialize the first time, then down again as he finally materializes.
The color of the Enterprise's phaser beams differ between episodes. In some they are blue, while in others the are red, yellow, or orange. The animation for the photon torpedoes also changes from a red-orange color to the more-often-seen white-colored animation.