We (among other things I'm a historian who does research in Japanese religion) generally don't refer to this conflict by a specific term as Japanese scholars do; I put "dispute over Buddhism('s adoption)" there merely as an example of the kinds of phrase that get used to discuss this situation. The term refers to the proxy battle being fought in the late 6th century between the increasingly (and eventually) dominant Soga clan, who favored Buddhism's adoption for various reasons both magico-religious and practical, and their various enemies at the imperial court at the time, who did not. I say "proxy battle" because this was just another aspect of the struggle for power and influence that the Soga and the other noble families were fighting at the time; the problem of Buddhism's introduction was merely another aspect of that. The Wikipedia Japan link provided is a good start; the link I give below provides some more solid background in English from a prof who posted the lecture notes for his undergraduate Japanese history course.