The crane is a symbol of health, and origami cranes are often given to those who are sick. It's believed that if you fold 1,000 paper cranes, one for every year of the mystical crane's life, your wish will be granted by the crane for your dedication. In some stories, it is believed that the cranes must be completed within one year and they must all be made by the person (or group of people) who will make the wish at the end. There is even a really beautiful, but sad, children's book about the myth called Sadako and the Thousand Paper Cranes by Eleanor Coerr.
A Daruma is a traditional luck charm rooted in Japanese folklore about a monk called Daruma, the founder of Zen Buddhism. A farmer made a doll based on the story so monks could bless them for luck. The dolls have eyebrows shaped to look like cranes, blank eyes and a beard that looks like two turtles. The original came only in red, but today there are different colors for different wishes. They are often sold in souvenir shops, and many businesses and politicians keep them for luck throughout the year.
The tradition of the paper crane comes from Japan's reverence of cranes in general. The Japanese people believe that they live for 1,000 years and represent happiness, longevity and good luck.
