Every Shinto Shrine I have visited in Japan has a place for these. What in the World is This?
The answer:
These are called ema. Visitors to Japanese shrines purchase these small wooden tiles from official vendors in front of the shrine. The ema are often decorated on the front with an image related to the shrine or to the animal associated with the Chinese calendar.
People then write a wish or prayer (onegai koto) on the back of the ema and hang them on stands near the shrine (as pictured here).
Most ema are shaped like what you see in the pictures that follow, though there can be some variation. A "standard" size ema, perhaps six inches wide by three inches tall and half an inch thick, costs about ¥500; some larger ema might cost as much as ¥1000.
You might be interested in more information about Shinto.
Here are a some examples of designs on the front of an ema:
Below are some examples of people's "onegai koto":
And this is an example of how the ema are hung near the temple:
