3rd Year JETs: The end of your contract also marks the end of the three year visa you receive at the beginning of JET. You must make a trip down to the closest Immigration bureau. You should double check the exact date by looking at the Period of Stay (Date of Expiration) on your Residence Card.
If you are a re-contracting 3rd Year JET, extend your period of stay before the end of your contract (you can apply up to three months prior although you will not be able to begin the process until your BOE finishes with the paperwork regarding your next contract). Your supervisor will help you with your paperwork to extend your visa.
If you are not re-contracting with JET and leaving Japan, you should switch to a 90 day Temporary Visitor Visa. This is especially crucial for not re-contracting 3rd year JETs as you cannot stay even one day past your visa, without likely having trouble at the airport.
Pro-tip: This now allows you access to the JR Rail Pass if you are planning on traveling.
If you are not re-contracting with JET and staying in Japan with different employment: You must inquire with your new employer and the Immigration Bureau of Japan about if you need to make changes to your visa.
If you are continuing as an ALT in a different company, it is likely you can keep your current Instructor visa if it doesn't expire with your contract.
If you are a third year JET who will be working as an ALT/instructor for a different company, you must ask your new employer for the paperwork to extend your period of stay.
If you are no longer teaching English, you will need to change your visa to reflect your new job.
Find your closest immigration bureau and their operational hours here (Gifu, Nagoya, Toyama)