I remember this episode. This is definitely my all-time favorite Trek episode with the Tribbles in it. The Enterprise is ordered to guard several tons of "wheat" onboard a space station.
Anyways, it involves a peculiar individual whom in turn gives Dr. McKoy a furry little critter to keep.
After several days on the Enterprise, the Tribbles multiply in the thousands and eat up the grain which the crew of the Enterprise must protect.
Kirk finds out why there are so many Tribbles on his ship later on and quickly investigates the matter.
In the end, they go back to the guy who had the first Tribble and force him to clean up the dead Tribbles, or be sent off to jail.