Attractive Bent Gender: Though it's not really an outright gender bend -Natsume just gets possessed by a Youkai who happens to be female, and who occasionally slips to the surface causing him to to exhibit feminine qualities (sometimes seemingly having longer hair). All of his friends start finding themselves disturbingly attracted to him.
Bishōnen: Looks almost exactly like his grandmother.
Chronic Hero Syndrome: Natsume will rush headlong toward Youkai, and all kinds of other danger if he knows someone is in trouble. Knowing full well that his opponents are more often than not much much stronger than.
Classical Antihero: Natsume. He's an ordinary human minus the ability to see youkai, and is filled with self-doubt and trust issues.
Cuteness Proximity: A mild example, but Nyanko-sensei calls him out on it once, saying he's far too nice to youkai he thought were cute.
Did You Just Punch Out Cthulhu?: He frequently knocks out strong youkai with a punch, the most frequent victim being Madara. It's useless against humans though, since it's his Soul Power which does all the work, rather than his physical strength.
Distressed Dude: Natsume has the bad habit of getting captured frequently. For instance, in episode 5 of season three, he gets captured by an ayakashi that's intent on cutting off his arm so that it can use it to retrieve its lost head. Natsume's friends are left frantically searching the house for him. He's also been kidnapped by Matoba two or three times. Nyanko-sensei usually bails him out.
Heroic Self-Deprecation: He considers himself an inconvenience to the people around him, and considers even simply asking for help with something to be a selfish indulgence.
Identical Grandson: Natsume Takashi looks a lot like his grandmother Reiko, original owner of the Book of Friends. and since many (if not all, Hinoe certainly seems to express a preference) Youkai don't really care too much about gender, Natsume tends to get Youkai who are looking for his grandmother on his tail.
Taken to it's logical conclusion when someone calls Natsume "Girly Man".
Orphan's Ordeal: A good bit of Natsume's difficulty with people (and his reluctance to tell the Fujiwaras about his abilities) stems from the fact that, up until then, he was passed around from relative to relative because nobody wanted to deal with the Creepy Child that kept telling "lies" about seeing ghosts and Youkai.
Abusive Parents: Chapter 31 confirms that several of the parents he's lived with were physically or emotionally abusive.
Parental Abandonment: Natsume's parents are dead, and every single family he's lived with before the Fujiwara's looked forward to the day he left their family, since he could see spirits.
Silent Scapegoat: Natsume often wound up as one in his adolescence, when Ayakashi started causing acts of vandalism around him; his options were either to let everyone believe he was some sort of delinquent, or let them believe he was some sort of crazy delinquent.
Jerk with a Heart of Gold: For all his claims to be a ruthless, cruel and uncaring monster who threatens to eat other youkai or humans on a whim, Madara is terribly benign compared to many other youkai that appears in the series. That image stayed for a little while to begin with, but is completely shattered when Madara pulls a Ghibli-style rescue on a little girl who fell down a cliff. Wanting the Yuujinchou was definitely one of his most important motives, and even this is called into question when Madara tempted Nastume to drink mermaid blood. Nastume called him out on this, saying if he was immortal, Madara would never get the Yuujinchou, to which Madara lapses into silence.
Giving us such gems as Nyan-Nyan Sensei, Nyankichi-kun, and Nyangoro. Tanuma eschews trying to remember it altogether, and just calls him Ponta.
Noble Demon: Madara continues to insist that he's really nasty and uncaring and only hangs around Natsume because the latter is his prey (or alternatively his servant). He's not really fooling anyone.
Not so Above It All: Nyanko-sensei regularly brings up his truly awesome powers and the fact that he is "not a fortune cat, damn it!" - until Natsume starts petting him, or hangs a toy in front of his face. Or he spots a grasshopper. Or a little girl has fallen down a cliff and needs rescuing.
Those Two Guys: Along with Nishimura, though contrary to the trope, they have fairly important roles in the plot as one of the few (non-psychic) people who are close friends of Natsume. They also get more screentime than anyone other than Natsume and Nyanko-sensei and a fair bit of fleshing out in the manga.
Cuteness Proximity: Although her eye for cuteness is called into question when she has this reaction to Nyanko-sensei (who everyone else regards as the cat equivalent to a Gonk).
The Quiet One: At first, but that was only because an evil youkai cursed her, so that the last thirteen people whose names she says will die. Natsume helps her break it. After that, she became more like the Genki Girl.
Blackmail: He likes to exploit Natsume's weaknesses, like his concern for a youkai's well-being or desire to keep his foster parents in the dark about his ability to see youkai because it might distress them, to force Natsume to collaborate with him. Several times.
Manipulative Bastard: He manipulates Natsume into helping the Matoba Clan expose an exorcist possessed by a spirit that hates the Matoba Clan and attacking them by conveniently leaving out exactly who was being attacked. See also Blackmail above.
Precocious Crush: Maybe, but he probably just really, really, really admires and looks up to Natsume as a Big Brother Mentor. The rock God's "humans, ayakashi, and beasts have different times" speech to the fox child in season three about why he and Natsume shouldn't be together and his resulting angst seemed to imply something else though.