This is a requirement of the immigration office for certain nationalities, but some language schools extend the list to include other nationalities for their own protection - overstays can seriously affect the reputation of a school, the school's relationship with the immigration office, and the way that the immigration office handles applications for visas through that school. The better the relationship, the easier it is for that school to get visas approved and the more jealously they try to preserve that relationship.
There are two kinds of guarantor: personal (as outlined above and required by the immigration office) and financial. You only need a financial guarantor if you cannot pay for your studies yourself. If you cannot provide proof of sufficient funds to cover your studies, then you may require a financial guarantor.
Students with Monbusho scholarships will not require a guarantor.
Personal Guarantors
A personal guarantor ("hoshounin") is someone living in Japan, either Japanese or non-Japanese, who is legally resident in Japan, and is employed full-time. Students cannot act as guarantors, nor can those who will not demonstrably be in Japan for the entire period of your stay in the country.
A personal guarantor is legally, morally and financially responsible for you. Even if you also have a financial guarantor abroad, your personal guarantor is in the country and hence still responsible for you in the event that you end up in financial difficulties. For this reason, the Immigration Office only allow those who are clearly capable of assisting you in case of difficulty to become guarantors.
Letter of Guarantee
This also means that the responsibilities of the personal guarantor are quite significant. Do not be surprised if people are unwilling to be your guarantor if they don't know you well - it is a lot to ask of an acquaintance. The guarantor will have to provide a letter of guarantee similar to this Immigration Office form, which is either sent to the school with your application or taken to the Japanese Consulate, depending on the visa you apply for. They will also be contacted by the Immigration Office to ascertain whether they are eligible to be a guarantor and their reasons for doing so.
Letter of Invitation
Similar to the form required from a guarantor, nationals of many countries will require a letter of invitation to Japan. This is from someone, again either Japanese or residing legally in Japan, who invites you to visit them in Japan. This is generally for applying for a tourist visa, and should in that case be taken to the Japanese Consulate when you apply for your visa. An example letter is linked here. See the next page (being a guarantor) for translations of the letter of invitation and letter of guarantee.
Financial Guarantors
If your personal guarantor in Japan is not going to guarantee your finances, then you will have to do so yourself or get someone else to do so. A financial guarantor can be anywhere - they do not have to be Japanese or in Japan - and can also be a company or organisation.
A financial guarantor will have to provide proof of savings (the amount will depend on what visa you are applying for and how long you intend to stay in Japan). They will also have to provide proof of relationship to you: either birth/marriage certificates or other official documentation to establish the relationship between you, or a letter from a public official (notary, judge, etc.) confirming that the relationship between you and the guarantor is as stated. All documentation must be originals or certified copies. If the Financial Guarantor is not related to you, then it may help to include a letter explaining why they are supporting you.
Proof of savings can be bank statements, letters from a bank or other savings body, and can be in the form of cash deposits or sometimes stocks and bonds, pension funds, etc. Any letters should state clearly the amount in the account, or state that over a certain amount is possessed by the account holder. Statements such as "has sufficient funds" or "is capable of supporting Mr. X in his studies" are not sufficient and will be rejected. Statements of the form "has a total of over 10,000USD in her accounts" are acceptable.
You should provide a signed letter from the financial guarantor stating that they are willing to act as such (see example) and the documentation proving their savings, plus proof of relationship if asked for to the school you apply through, or directly to the Japanese Consulate if requested to do so.