Types of visa
Temporary visitor or "Tourist Visa"
If you arrive without visa in Japan, you will be issued a short-stay visa. For most Europeans it is valid for 90 days. This visa does not permit you to work in Japan.
If you are intending to study more than a few weeks/months in Japan, you should ask for a student visa. There are several types (pre-college, college...), but they are generally valid for one year. They are expensive as you must pay the school fee for a full year in a recognised establishment. The college visa entitle you to work part-time (up to twenty hours per week). You must get a re-entry visa if you leave the country and come back, otherwise you lose your visa. Student visa holder can only buy a single re-entry visa (3000 yen), which means they have to get a new one everytime they exit the country.
Available to anyone who is studying something typically Japanese, such as martial arts, ikebana or calligraphy. It allows to work part-time (up to twenty hours per week), is valid for one year and renewable. It is interesting for people who intend to stay a long time in Japan without real intention to find a serious job.
Only a few countries have agreement with Japan regarding this visa. Among them are the UK, France, Germany, Canada, Australia, New Zealand and South Korea. You must be under thirty years old to be eligible. This visa is valid for one year and allows you to work. The difference with a working visa is that you don't need a sponsoring company, nor to have a company ready to engage you at all. You don't need a university degree neither. The number of application is limited per year.
They are valid for one year. You will need a university degree, be over 18 years old and have a sponsoring company. You need a re-entry visa to exit and re-enter the country. Working and spouse visa holder can buy the "multiple re-entry visa" (6000 yen) valid until the end of their (working/spouse) visa.
If you are married to a Japanese national, you are eligible for this visa. It allows you to work and is easily renewable without any other condition. You still need a (multiple) re-entry visa to leave the country without losing your visa.
Available to those who set up a company or branch in Japan. There are certain conditions (e.g. employing at least two Japanese nationals full-time). However, it is now possible to start a company with a capital of 1 yen.
It allows you unlimited stay in Japan and basically the same rights as Japanese people. You'll need this visa if you want to buy a house or get a loan from the bank. It of course allows you to work. To be eligible, you will have to stay (legally) at least five years in Japan, speak some Japanese (you don't need to be perfectly fluent) and have a full-time job. It's easier to obtain if you already have a spouse visa - that is if your husband or wife is Japanese. You might wait one year to get this visa.
Note that there are other kinds of visas, such as diplomatic, official, artistic, religious, medical, legal, researcher, entertainer, skilled labor, etc.
For more information about Japanese visa, check the Japanese Ministry of Foreign Affairs's website.
How can I obtain a visa ?
If you arrive in Japan without visa, you will automatically be given a 90 days temporary visitor visa. Normally all other visas should be obtained before coming to Japan. However, it is now usually possible to change a temporary visa into a working, cultural or spouse visa, by applying at an immigration office in Japan.
How can I extend my visa ?
Temporary visitor or "Tourist Visa"
British, Irish, German, Swiss, Austrian and Mexican citizens can extend it for three more months. For others, it may be quite difficult to get an extension. It will be necessary to have a good reason and show you have enough money to support yourself. Even so, you might not be granted more than a few aditional weeks.
Student, Cultural, Working & Spouse Visa
These visas can be easily extended as long as you are still studying/working/married. Just go to the immigration office with a proof that you are still eligible.
For more information about visas and immigration, you can consult the Immigration Information Center in Tokyo.