My employer wants to send me to work in U.S. with L-1B visa at its office near Boston. To meet the affiliation relationship of the L visa, what is the exact definition of  "branch office" for the L visa application?

A "branch" is an operating division or office of the same organization housed in a different location. While the branch office or division may be distinctly organized or incorporated in the U.S. or the foreign country, it is necessarily 100% funded, controlled by, and forms an integral part of, the company to which it belongs. 

Likewise, the U.S. company may be a branch company of the foreign company, or the foreign company may be a branch company of the U.S. company.