A gaijin house (literally: foreigner house) is a place where multiple foreigners can live for up to a month. If your trip to Japan is going to be brief, this is an economical option. You will be living with other foreigners, so you will have a chance to meet people who are in the same situation as you. Your room-mates could turn into good friends, or at the very least, people with whom you can communicate. Most gaijin houses are located in Tokyo (so are most foreigners) and they are operated by a mixture of real estate agents and private owners. The primary advantage of gaijin houses is that they are cheap.
Manshon (mansions) are the tall high-rise apartment buildings common throughout Japan. For all intents and purposes, weekly manshon are the same as their residential cousins. The difference is that they are fully furnished.
This is an excellent option for those who are in Japan only temporarily or are searching for a place to live, but want more privacy than that offered by a gaijin house. The rent is usually listed per day (in spite of the name), and in some cases these manshon may be rented for longer than a month.
Staying with a Japanese host family
This option is primarily available to students, who are almost always able to arrange for a placement through their university. The advantages of staying with a host family are the low cost and integration into Japanese family culture, not to mention delicious home-cooked meals. If you and your family are a good match you will learn a lot.
Staying with a host family limits your independence, however, as you must be considerate of the household´s needs (especially if there are children). This means sharing the limited space and keeping reasonable hours. If you are someone who spends a lot of late nights on the town you should save yourself and a Japanese family a lot of grief by getting a place of your own.