People who wish to live permanently in Japan must either obtain a Permanent Residency Permit (eiju kyoka) or become a naturalized Japanese citizen (kika).

Permanent Resident Permit (Eiju Kyoka)
With a Permanent Residency Permit an individual may live permanently in Japan as a foreign national. If you wish to apply for a Permanent Resident Permit, enquire at the immigration bureau for your district.

Naturalization (Kika)
Naturalization (kika) means a foreign national forfeits his/her foreign nationality and obtains a Japanese nationality.
To become a naturalized Japanese citizen, a permit form the Minister of Justice is necessary, so you should enquire at the Regional Legal Affairs Bureau (Chiho Homu Kyoku) for your district.
Your name will be registered in kanji (Chinese style letters), hiragana or katakana.

This information is from the CLAIR website. The Council of Local Authorities for International Relations (CLAIR) was established in July 1988 in response to rising concerns about local level internationalization in Japan.