An inauguration ceremony was an event in which a politician was installed.
In 2267, the USS Enterprise was expected to be present for the inauguration of Altair VI's new president, an occurrence which was expected to help stabilize the Altair system. The inauguration ceremonies were re-scheduled for seven solar days earlier than previously set.
Captain James T. Kirk was made aware of this fact via a message sent by Admiral Komack in which he ordered the Enterprise to alter their flight plan so they could attend. Upon Kirk's request for a day's delay (to transport Spock to Vulcan, although the admiral wasn't informed of this), Komack refused, explaining why the inauguration was so important.
After Komack ended the call, Kirk decided to head to Vulcan anyway, explaining to a confused Leonard McCoy that the Enterprise would have been one of three starships present, something which would be more for show rather than necessity.
Kirk's request was ultimately granted after the fact due to T'Pau's intervention. Following his and Spock's return to the Enterprise, he ordered Pavel Chekov to set course for Altair VI. (TOS: "Amok Time")