A generation was a group of related individuals all belonging to a specific time period. The term could also refer to the time period itself. Some technology was also referred to as belonging to a generation; that is, a certain number of designs removed from the original.
At least six generations of Gary Mitchell's maternal ancestors were said to have dabbled in metaphysics, according to his medical record. (TOS: "Where No Man Has Gone Before")
As of 2267, there had been many generations of Horta living on Janus VI. (TOS: "The Devil in the Dark")
In 2268, Spock estimated that "assuming one tribble, multiplying with an average litter of ten, producing a new generation every twelve hours over a period of three days", the total amount of tribbles on Deep Space Station K-7 now amounted to "one million seven hundred seventy one thousand five hundred sixty one", also taking into consideration how much quadrotriticale they had eaten and the volume of the storage compartment they were inside. (TOS: "The Trouble with Tribbles")