This is Natsume Yūjin-Chō, a truly excellent anime. This rec is for the first episode only, and I have seen up to the third episode only. However, these three episodes are brilliant in and of themselves. This is a short story series - there tends to be one story per episode, with a small amount of over-reaching plot.
The premise is that Natsume, our main character, can see spirits and has seen them ever since he was born. No one else can see them, and so no one believes him when he sees them. However, his grandmother could see them - but she has passed on.
There is more to this story...
This is Natsume. He's our main character, and he mostly keeps the story going. I like him, but you may not. Still, give him a try - he grows on you.
Also, he can see spirits!
Remember the cat in the first image? This is his alternate form. He is AWESOME and I can't remember his name. But he's funny, cool and awesome all in one.
Okay! Those are your two main characters. The stories are short, sweet, and never feel rushed or anything. This is anime haiku, practically. Enjoy it!
Two more pictures from the first episode, then a download link:
Natsume's grandmother. She was awesome.
One of the spirits. :)
This anime has sweet moments, sad moments, and funny moments. Mostly, though, there's a feeling of peace. I love that.
Okay! There are thirteen episodes in the first season, and another thirteen episode for the second season.
Download link to the first episode:
Ask for more episodes in the comments, please.