(covers information from several alternate timelines)
Initials were the alphabet letters which began one's first, middle, and/or last names. Some people, when identifying themselves to each other, consistently included their middle initials, such as James T. Kirk.
Initials were sometimes signed on documents (in lieu of one's full name) to indicate that one had read and acknowledged them. In 2267, while testifying at Captain Kirk's competency hearing, Lieutenant Nyota Uhura was shown a bulletin from Starfleet Command that had hers and the captain's initials at the bottom. She confirmed that fact when asked to by Spock. (TOS: "The Deadly Years")
Carving one's initials into a tree was considered a sign of affection. While attending Starfleet Academy, Jean-Luc Picard carved the initials of A.F. into Boothby's favorite elm tree. Wesley Crusher mentioned that Boothby had told him this in 2368.(TNG: "The Game")
In 2372, the Kathryn Janeway from a duplicate of USS Voyager told her starship's Samantha Wildman that she had been making a blanket for Samantha's baby, and needed to know the child's initial so she could add it to the blanket. (VOY: "Deadlock")
In an alternate timeline 2374, while playing a trivia game in an inoperable turbolift, Harry Kim told B'Elanna Torres that she was probably confusing Cary Grant with Clark Gable as the male lead in To Catch a Thief , as they both had the same initials. (VOY: "Year of Hell")