Abusive Parents: Some of his previous foster parents 'got drunk and hit him', and others 'forgot' to feed him. Natsume dismally blames himself for their behavior. They always started out nice.
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.
Soul Power: He has a lot of spiritual power, which is the reason behind all of the other supernatural tropes that apply to him, including his ability to punch youkai clear across a room.
Nishimura:It felt like that was the first time I saw the ever smiling transfer student smile. But when I looked at it another way, I thought to myself that his normal smiles must all be fake.
Beware the Silly Ones: Calls himself middle-aged, acts accordingly, and looks nothing near capable of kicking butt, but get him angry and he turns into a...
Big Badass Wolf: His true form is a huge and rather powerful inugami.
Gonk: Pretty much known to everyone as "Natsume's hideous cat".
Ugly Cute: Taki (and probably you) finds him to be irresistibly adorable, and glomps him whenever she gets the chance.
I Am Big Boned: Very insistent that his cat form is not fat, merely very fluffy.
If You Die I Call Your Stuff: His relationship with Natsume is formed on this trope - in return for serving as his bodyguard, he gets the Book of Friends after Natsume dies.
Intellectual Animal: Kind of, seeing as most Youkai tend to be at least sentient.
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.
My Name Is Not Durwood: Nobody besides Natsume bothers to get Nyanko-sensei's name right.
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.
Badass: You know you're one when you can defeat every single ayakashi you come across. Two bullies throw a rock at her and she manages to knock it back with a stick, without flinching or hesitation.
Broken Bird: Became rather angry and violent thanks to the treatment she's received over the years.
Even the Girls Want Her: Hinoe's attraction to her is anything but subtle, though slightly subverted as Hinoe Does Not Like Men and most non-humans tend to disregard gender.
Glory Seeker: Probably the reason why she takes the names of the youkai.
Posthumous Character: Reiko. Once Natsume has returned the name of a youkai, a flashback is usually shown about the circumstances surrounding when Reiko first took the name.
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.
Not Good with People: Tanuma also sees Youkai, but worries that he won't be of any use, or will just be a burden on Natsume, because of his weak power - he can only see shadows and glimpses of Youkai.
Psychic Powers: Can understand Nyanko-sensei, feel the youkai's presence and can sometimes see shadows, can't exactly see their corporeal forms.
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.
Adaptation Expansion: In the original manga (chapter 5) she moved away after the test of courage. In the anime she is still interacting with Natsume after the test of courage.
Secret Keeper: Promises not to tell anyone about Natsume's ability to see youkai(although he refuses to admit he can, she's just convinced of it and is extremely nosy on her own).
Sugar and Ice Personality: Is rather strict even to her own classmates, but actually rather sensitive on the inside.
Distinguishing Mark: Has a lizard shaped birthmark that moves across his body and can only be seen by other spiritually inclined people. It's actually an ayakashi living inside his skin.
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.
Eye Scream: Above mentioned Eyepatch of Power also covers up a wound left from an ayakashi who attacked Matoba's right eye. He still has right eye and can see but cannot estimate distances well.
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.
Well-Intentioned Extremist: Constantly insists to Natsume that what the Matoba clan does to youkai is necessary for the good of humans, even if they end up hated by humans by themselves.
Contrary to the stereotype, however, he cannot transform (possibly because he is just too young) and is rather straightforward, not tricky in the least, nor particularly cunning.
Petting Zoo People: Although ordinary people just see him as a regular fox.
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.