The entrance process for international students is different to that for Japanese nationals, with around 95% of national, 65% of public universities, and 45% of private universities requiring foreign undergraduate students to take the Examination for Japanese University Admission for International Students (EJU). The examination tests Japanese language ability, as well as knowledge on other subjects such as mathematics, chemistry, and “Japan and the World”—the exact subjects depending on the university and course to wish the student will apply. It can be taken in Japanese or English, and because it also tests Japanese language ability most universities do not require a JLPT score. The EJU is held twice per year (June and November) and can also be taken in selected Asian cities outside of Japan.
Universities which do not use the EJU (or require a separate entrance examination as well) will usually set a different entrance examination for foreign students which tends to be easier and shorter (and you may also be able to take it in English or another language). The EJU does not apply for those hoping to enter graduate school. Here the JLPT will be required along with an entrance examination set by the university (which in many cases can be taken in English).
Incidentally, the university application process for Japanese nationals is briefly explained here.