The official name of the “My Number”, as it is called in Japan, will be 個人番号(Individual Numbers, or “The Social Security and Tax Number System”).
The system will (in theory) simplify issues related to tax, national pension, unemployment and medical insurance etc. in giving the person one unified number for all purposes. Private businesses will also handle Individual Numbers for social insurance and withholding tax.
All foreigners registered in Japan will be covered by the system, as well as Japanese citizens
The number will be used to better share information among the various national and local authorities (start January 2017 at national level and July 2017 for local authorities).
A notification will be sent out in October 2015 including your actual number.
Application for the actual card can be done in January 2016. There is no must to always be carrying the card as with the Residence card/Alien registration card. The Residence card/Alien registration card cannot be used as a substitute, nor will the numbers be the same.
More information can be found here (the webpage of the Cabinet Secretariat).