Introduction
Wasan is the name for the traditional stye of mathematics that was developed in Japan before the Mejii Period. Much of the early development of Wasan was borrowed from the mathematics formulated in China, and was motivated by astronomical calculation. During the Edo period, Wasan underwent a revolution at the hands of Seki Kowa. Kowa and his followers developed a distinctive style of mathematics, and formulated concepts being developed at the same time in the West, such as linear algebra and calculus. Another feature of Wasan is Sangaku, which were geometric puzzles that were carved on temples without a solution. When contact with the West was established in the Mejii period, Wasan was abandoned in favor of western mathematics. Wasan and the history of Japanese mathematics has been little studied in the west.
Early Wasan
Early Japanese mathematics was essentially imported from China. It included knowledge of geometry, and Japanese mathematicians made use of the Pythagorean Theorem to turn geometric problems into algebraic ones. Another motivation of Japanese mathematics was the creation of an astronomically accurate calendar. There are not many documents concerning mathematics surviving from this time.
Around the 13th century, the Japanese version of the abacus, the soroban was developed to perform large, tedious calculations very quickly and efficiently. One can learn to perform calculations extremely quickly with the soroban. Even today, some Japanese students learn to do calculations mentally by visualizing a soroban in the air.

Seki Kowa (Seki Takakazu) 16421708
Seki Kowa was the most influential Japanese mathematician before contact with the west, and it was he who established a school giving Wasan its flavor distinct from both Western and Chinese Mathematics. Seki Kowa achieved major accomplishments, paralleling mathematical developments that were taking place in Europe at the same time period. Seki Kowa was born in 1642 to a Samurai family, but for unknown reasons was adopted into the Seki family. Little is known about his personal life and history, but his family was either from Fujioka or Edo. Most historians believe that he was self educated, since most of his work is startlingly original. His major results were the discovery of the determinant, a mathematical construction used to solve systems of linear equations by eliminating dependent variables. This is rather amazing, since the determinant would not be discovered in the west for another 12 years by Gottfried Leibniz! Another major discovery attributed to him is yenri, or the circle principle. This is essentially equivalent to the calculus that was being developed by his contemporaries Newton and Leibniz in Europe, and is so called because its first application was the computation of the value of pi to 15 decimal places. 
Sangaku
Sangaku refers to the Japanese practice of carving mathematical problems and theorems on wooden tablets, which were then displayed in public places, often in temples. These problems are often referred to in English as "Japanese Temple Problems". They were usually of a geometric nature, and involved finding areaa, volumes, and radii of circles. Most of these tablets were sadly lost during the modernization of Japan in Mejii Period.
Find the Volumes and Surface Area!
What are the radii of the blue circles?
A Proof of the Pythagorean Theorem Inscribed Circle Theorem
What's the volume of the tetrahedron? Larger Sangaku Tablet
Sangaku were puzzles that many practiced, including nobility, priests, and commoners. There are many books containing these problems with english translation. There is a wonderful website http://faculty.matcmadison.edu/kmirus/Reference/SanGaku.html that contains these problems online.
The Difference Between Wasan and Western Mathematics
In light of the fact that the Japanese developed many mathematical concepts such as calculus and linear algebra at the same time and even before mathematicians in Europe, why did the Japanese abandon Wasan upon contact with the West? In what sense was western mathematics "superior" to Wasan?
I believe one of the primary differences between Japanese mathematics and Western mathematics was the context in which they were developed. For example, it is true that Seki Kowa and his followers developed a sophisticated equivalent to the calculus of Newton and Leibniz, but they applied it only to geometrical investigations, such as finding areas and volumes. Newton developed calculus for the purpose of describing a theory of gravity. In the west, this to the development of physics, and from this point onward, much of the development of mathematics was motivated by questions arising in physics. Many questions in physics are concerned with forces and quantities that are not easily visualized such as energy and momentum. Thus a new level of abstraction of mathematics arose in the west, while in Japan, mathematics remained focused on solving geometric problems that could be easily visualized and were aesthetically pleasing. The higher level of abstraction in mathematics gave it greater problem solving power and opened new avenues for mathematical exploration. It is for this reason I believe that although Wasan and Western Mathematics were on an equal footing at the turn of the 18th century, Western mathematics accelerated its development and was much stronger than the mathematics of Japan at the dawn of Japanese modernization.
Links to websites:
Sangaku Problems: http://faculty.matcmadison.edu/kmirus/Reference/SanGaku.html
Online version of book "History of Japanese Mathematics": http://www.archive.org/details/historyofjapanes00smituoft
Wikipedia article on Wasan: http://en.wikipedia.org/wiki/Japanese_mathematics
Wikipedia article on Seki Kowa: http://en.wikipedia.org/wiki/Seki_Kowa