Firstly, you have to turn in financial statement, which shows the capability to pay tuition fee and have enough money to spend daily life in Japan. Tuition depends on the university you like to go. If it is a public university, the tuition per year is around 500,000 Japanese yen, whereas if private, it is about 800,000 Japanese yen. In addition to that, you have to pay an entrance fee, which is between 200,000 and 300,000 Japanese yen. Living expense varies depending on where you live. For example, the monthly rent for a studio in Tokyo is about 80,000 Japanese yen, which is twice as much as that in other local cities. Of course you have to take some scholarships and student loan into consideration.
Japanese language requirement
Japanese proficiency is mandatory. But, its required level changes depending on the university you like to go to. There are two Japanese proficiency test, JLPT, Japan Language Proficiency Test, and English section of EJU, Examination for Japanese University Admission for International Students. JLPT measures your listening and reading skill only. The JLPT is offered in five levels, N1 is the most difficult, on the other hand, N5 is the easiest. N1 level proficiency is required in some famous university, basically. EJU examines not only Japanese proficiency, but also knowledge of other subjects, such as mathematics and social science. In the Japanese section, there are listening section, reading section, and essay section. Universities require prospective students to turn in the either score, JLPT or EJU.
Other requirements for undergraduate
Some departments of famous universities require English proficiency to students whose mother language is not English because English is used in some class. Basically, you have to turn in TOEFL score. Probably 80 on iBT is enough for entering such a university. Also other departments require students to turn in the score of Mathematics or Science section of EJU. And there is a short interview, which probably measures the oral communication skill and motivation for going to university.
Requirement for going to graduate school
If you like to go to graduate school in Japan, the examination is different form that for undergraduate school. Firstly, there is an exam of the subject you major in. Also, there is an English exam, most of which is composed by translation problems from English to Japanese. The last one is an interview about your research. Before the interview, you have to turn in your research proposal, then some professors ask you about your research or the reason why you like to go to graduate school. Of course, in addition of these, you have to turn in your JLPT or EJU score.