Japanese animation as a medium is known to tackle many genres that animation from the United States generally steers clear of. In addition to family comedies and children's series, anime also delves into more mature genres like drama and horror. But anime can also tread ground mostly unfamiliar to those of us in the West. One genre that seems to be particularly unfamiliar is called iyashikei.
The term iyashikei comes from the root verb iyasu, which has various meanings including "to heal" or "to relieve (pain)," and refers to anime, manga and other things (including idols and comedians) which embody these properties. There are very few anime and manga titles which genuinely and completely fall into this genre; the classification is rather subjective based on personal experience and there is a lot of crossover into other genres. In short, your mileage may vary. The genre also seems to be fairly unpopular in the West because it goes against the common expectation that anime be flashy, bombastic and action-oriented, but I tend to like these shows for their simple, relaxing qualities. Here are some examples of titles which are either generally considered to fit well within the iyashikei genre, or which strike me as crossing into that genre from time-to-time:
Aria The Animation, Aria the Natural, Aria the Origination
This series takes place in the future after the terraforming of Mars into a beautiful, water-filled planet called Aqua. In the city of New Venice (modeled after, what else? The city of Venice in Italy), a group of young women strive to become "undines," gondolier tour-guides. The story focuses on the Aria company, the smallest tour business with only two employees.
The story is very slice-of-life, and concentrates on the relationships between the undine trainees as they're set against the beautiful, familiar but slightly alien backdrop of New Venice.
Yokohama Kaidashi Kikou
This manga and 2 episode OVA take place in the distant future after an apocalyptic even which has changed the topography of the world and left humanity to fall into a decline. The star of the story is Alpha, a female android who runs a coffee shop while her master is away.
The events surrounding the apocalypse aren't ever shown; the story is more of a meditation on the beauty that can be found in the changed world and the melancholy of humanity's slow decline and disappearance.
Bartender
Hidden in a back alley in Ginza is Eden Hall, a quiet bar run by genius bartender Sasakura Ryuu. As the series unfolds, many people, each from different backgrounds and different stations in life find themselves in Eden Hall, where Sasakura helps to ease their troubles by mixing them the perfect cocktail.
The series is definitely unusual due to its lack of over-arching narrative and very few recurring characters, but its refined atmosphere and Sasakura's attention to detail makes this series pleasant to watch and even educational (the recipe for the main drink featured each episode is provided during the end credits).
Besides these three titles, the slice-of-life series Sketchbook: Full Color's and the bishounen-centric Miracle Train are often put into this genre as well.
In addition to the few series which can truly be called iyashikei, there are some series that, from time-to-time, do take some tentative dips into the pool. Here are a couple series that have their iyashikei moments:
Natsume Yuujinchou
While this show, about a boy who can see spirits, is more often classified as a shoujo or supernatural series, the attention paid to the details of the natural world that surrounds him is often exquisite. The passage of the seasons is subtle, but the beauty of the fallen leaves or the crunch of the newly-fallen snow adds beauty to the rural setting.
This is one of the rare series which, though often comedic and action-oriented, also uses silence to great effect and conveys a certain sense of wonder to the viewer.
Mushi-shi
Though the show can admittedly be as unsettling as it is relaxing at times, overall while watching it I was graced with a quiet feeling of melancholy.
The main character, Ginko, travels the land to document mushi, small creatures which are neither plant or animal but which often have magical or supernatural properties. When mushi and human meet, the results may be pleasant or magical, but can often be harmful or tragic as well.
There's a certain serenity in the scenes where Ginko travels, and nature is portrayed beautifully.
Iyashikei is as difficult to concretely define as shows that fit the description are to locate, but hopefully this is a start. Which series offer a healing effect to the viewer are very subjective as well, so one person's list may not match another's.
If you want more information, Ask John at Animenation has addressed questions regarding iyashikei more than once:
In the third volume of the Journal Mechademia, which concentrates on Anime and Manga, there is a short review of Yokohama Kaidashi Kikou (the volume is available to purchase from Amazon, and it's well worth it, as are the other volumes).
Mushi-shi has a lot to offer other than a relaxing atmosphere, IMO. It's pretty challenging and thought-provoking in the long run, and I also admired the live-action version of the story (filmed by Katsuhiro Otomo of "Akira" fame).
I'm not really very familiar with that term, but I have seen others who say that "Someday's Dreamers" is also in that genre.