Individuality or self-identity was the concept of being or living a separate or distinct existence. Individuality, or being an individual or person was about being autonomous and independent. (VOY: "The Gift")
According to Jean-Luc Picard, "the Kurlan civilization believed that an individual was a community of individuals. Inside us are many voices, each with its own desires, its own style, its own view of the world." (TNG: "The Chase")
According to computer Landru, when one was absorbed in "the Body", their "individuality will merge into the unity of the good, and in your submergence into the common being of the Body, you will find contentment and fulfillment. You will experience the absolute good." (TOS: "The Return of the Archons")
James T. Kirk, John Doe, and Kurn all experienced a loss of self-identity when the were awoken for the first time after experiencing amnesia, all immediately inquiring, 'who am I?' (TOS: "The Paradise Syndrome"; TNG: "Transfigurations"; DS9: "Sons of Mogh")
During the trial held to determine Data's status as a person or if he were the property of Starfleet in 2365, Captain Picard argued that self-awareness was a key ingredient of one being an individual. Commander Bruce Maddox indicated that what made someone self-aware was that they were conscious of their existence and actions, that was, "you are aware of yourself and your own ego." When Picard turned to Data and inquired directly, "what are you doing now?" Data responded that he was "taking part in a legal hearing to determine my rights and status. Am I a person or property?" When Picard inquired whether or not Data knew what was at stake, Data replied, "my right to choose. Perhaps my very life." Picard echoed Data's response, "'My rights.' 'My status.' 'My right to choose.' 'My life.' It seems reasonably self-aware to me." (TNG: "The Measure Of A Man")
In 2368, Data observed that the Tamarian ego structure did not allow them to present themselves with a sense of self-identity. Rather, "their ability to abstract is highly unusual. They seem to communicate through narrative imagery by reference to the individuals and places which appear in their mytho-historical accounts." (TNG: "Darmok")
When Geordi La Forge attempted to explain individuality to Hugh, following his disconnection from the Borg Collective, he used the example, "every time you talk about yourself, you use the word "we". We want this, we want that. You don't even know how to think of yourself as a single individual. You don't say, I want this, or I am Hugh. We are all separate individuals. I am Geordi. I choose what I want to do with my life. I make decisions for myself. For somebody like me, losing that sense of individuality is almost worse than dying." Captain Jean-Luc Picard later considered that upon returning Hugh to the Collective, and "in that short time before they purge his memory, the sense of individuality which he has gained here might be transmitted throughout the entire Borg Collective. Every one of the Borg being given the opportunity to experience the feeling of singularity. Perhaps that's the most pernicious program of all. The knowledge of self being spread throughout the Collective, in that brief moment, might alter them forever." (TNG: "I Borg") It was later revealed that after Hugh was returned and interfaced with the other Borg, he transferred his sense of individuality to them and as a result, it nearly destroyed them. (TNG: "Descent, Part II") When Picard was later confronted with why he chose to make the decision to return Hugh to the Collective, he revealed that "when Hugh was separated from the Borg Collective he began to grow and to evolve into something other than an automaton. He became a person. When that happened, I felt I had no choice but to respect his rights as an individual." (TNG: "Descent")
When The Doctor experienced a radiation surge which caused his program to malfunction, he "experienced an elaborate delusion concerning the nature of my existence. Human or hologram. Person or projection. Why? Why would my program focus on such an esoteric dilemma?" Kes, then explained that she too, "sometimes ask[ed herself] those kinds of questions. 'Who am I?' 'What am I doing here?' 'What's my purpose in life?'" (VOY: "Projections")
According to Q, the death of Q "caused chaos and upheaval. Because even though he was gone, his calls for freedom and individualism continued to echo in the ears of those who believed in his teachings, myself among them. I sounded the trumpet and carried the banner. Naturally, others followed. The forces of the status quo tried to crush us once and for all, but we fought back. And now there's a cosmic struggle for supremacy, and the battle is spreading, causing hazardous repercussions throughout the galaxy." (VOY: "The Q and the Grey")
Years later, The Doctor, when referred to otherwise unseen significance of a cytoplasmic matrix they discovered, he explained that perhaps one day, it "may fall into the primordial sea of a distant world where it could ignite an evolutionary process. Eons from now, a creature not unlike you could emerge, look up at the stars and ask, 'who am I?' 'How did I come to be?' It's the miracle of creation, Seven. Doesn't that excite you?" (VOY: "Body and Soul")