The old standby for westerners trying to find work is the "English teacher" (French, German and Spanish are also popular languages here). So, the competition for English jobs is fierce and the pay is low. There are openings in elementary, and junior and senior high schools, but these require teaching certification and often a few year's experience. There are a lot of language schools here (Geo, GABA, Aeon, etc.) that will offer visa sponsorships, but most job openings go to holders of working holiday visas first; the positions are often part-time (maybe $100 or $200 per week); and some schools want ESL (English as a Second Language) experience.
There's also the JET program, which will sponsor people from different countries to live in Japan and teach English at local public schools, but the program is only for one year, and there are maybe only 50 positions for 5000+ applicants from the U.S. Your odds of getting into this program are low; everything revolves around how well you can complete in an essay writing contest.
This leaves private tutoring. There are two networking sites for students trying to find private teachers:
This is a free site for listing yourself as a language teacher. The dashboard shows you how many people have viewed your profile, and if someone wants to contact you, you'll get an e-mail notice. You are encouraged to upload a photo, and to keep your profile updated once a month. Recommended prices are 3000 yen per hour for individual students. In addition, findateacher has a monthly party at the Sonoma bar and grill in Shibuya (3000 yen = $30 USD), which is really a networking session for students wanting to talk to the teacher before requesting teaching. Findateacher charges the students an online finder fee for contacting teachers, which is why the parties are so popular (the parties give the students a way to bypass the online finder fee).
This is also a free to use service for listing yourself as a teacher. The service makes its money selling learning supplies (textbooks and flash cards), making it more popular with the students. Otherwise it's the same as findateacher.
As of September 8, I had 7 hits on my profile on findateacher, 30 hits on findastudent. And 0 requests to contact me for either site after 1 week. So, obviously, having a good profile is important, as is location and how much experience you've had teaching (if you lie about your experience, you'll be blacklisted from the site).
One advantage to private lesson teaching is that you can choose where to have the classes, such as at a coffee shop or library, and when. It offers flexibility to those of us that work part-time at a language school and need to make more money. The disadvantage is that you can't get guaranteed work. Having 5 students a week means getting no more than $600 per month.