Brian is a drug addict middle-aged man whom you encounter three times during the game.
Random characters are minor characters you can find around Liberty City. They are marked by a blue icon on the map, but only when you are nearby.
They provide you short missions, most of which are necessary for 100% completion of the game. After you meet all the characters, the achievement/trophy “No More Strangers” unlocks. If some characters die during the storyline, it doesn’t prevent you from achieving 100% completion.