Horses used to be considered as messengers of the gods. Those who could afford it, would bring a horse at the shrine with the hope of their wish to come true. Since horses were expensive, in time a wooden plaque with a horse drawn on it ended up being enough. Hence the name. And then, various patterns have appeared on such wish plaques.
The wishes range from success in business, to health, good result in an exam, love.
This is Murasaki Shikibu, who wrote "The Tale of Genji" (Genji Monogatari, 17th century):
厄 yaku is misfortune, bad luck. By leaving it at the shrine, one symbolically gets rid of bad luck.
開運 below means "better fortune":
Some ema I saw in Kyoto:
Few ema in Tohoku, Noh figures included:
And some ema from Umeda, Osaka - the shrine Ohatsu Tenjin:
It is interesting to see omikuji, fortune papers and another kind of making wishes in Japan, on an ema:
An Inari shrine represented on this ema, with kitsune, the fox:
The love story of Ohatsu and Tokubei - blessing people with a good relationship:
On the right side, in red, it is written 縁結び enmusubi, marriage:
Ohatsu and Tokubei are a symbol of eternal love.
And the latest wish, to have a beautiful face: 美人 bijin = beautiful person. Everybody draws a face and prays to become beautiful.
"A Prayer for Beauty: Physical beauty is a fragile flower. Inner beauty is a lifelong treasure. Wish for inner beauty":
The animals in the zodiac often appear on such plaques and they are sold according to the animal of the current year. Since we are in the year of the dragon, many dragons can be seen everywhere.
Below are such zodiac animals made by a very skilled neighbour - nice handicraft: