I stumbled upon this series entirely by accident and started in the middle of the series (5 I think) and had to go back and watch the whole thing. What a tremendously wonderful anime!
Natsume is a teen-aged boy who can see "youkai" (Japanese spirits/monsters...their form varies so much that it's a little hard for me grasp exactly what they are...perhaps someone more familiar with Japanese culture would). This ability has caused nothing but problems for him particularly given that he was orphaned at an early age.
At the beginning of the series, he's been taken in by a childless couple who seem to be the first family to treat him with kindness. We see him struggling to connect with the couple as well as other teen-agers who want to be his friend. While Natsume is shown to be compassionate, caring and responsible but he's also portrayed as emotionally damaged by his childhood experiences and a bit reckless at times. We see these characteristics played out time and time again via his interactions with youkai. Natsume is frightened by them but wants to help them as well given that many youkai come to him with problems to solve (generally involving the titular "Book of Friends"). What I find particularly interesting is how the writers of the series present understandable motivations for some of the most frightening youkai (though some are just scary for their own sake).
His "spirit guardian", a cat called "Nyanko-Sensei" who can transform into a terrifying monster if needed, finds his compassion and recklessness irritating and often scolds him for it. However, in one of the most subtle touches in the series, we see the ornery dismissive cat becoming more and more fond of Natsume (though he denies it strenuously).
The pace of the films is often slow-moving and sometimes there is very little plot. But each episode ends in a positive, life-affirming way. Some even bring tears to the eyes. Well worth watching for those weary of the action-packed violence of other shows in the genre.