A "Brian Moment" can be used in two different ways. It is typically added at the end of a sentence. It can define anything even slightly out of the ordinary, or weird (Similar to the "Only at Miller Grove" meme), or it can be used to describe when a teammate in a competitive sport/game does something counter-productive to your team. (ex. Scoring on your own net in soccer). If you see anything out of the ordinary happen around you, instead of ignoring it, please feel free to say "Brian Moment" to whatever happened.

Briefly, "the realization that each random passerby is living a life as vivid and complex as your own." Originally from the Dictionary of Obscure Sorrows, which has a lot more beautiful neologism definitions like this you might enjoy.