Basic Residence Registration Network System （住民基本台帳制度 Jyumin Kihon Daicho Seido/ in short 住基ネット Jyuki Net）starts for foreign residents in Japan from 8th July, 2013.
You don't really have to do anything about this system, but your number will be sent to your post around that date so please check out.
If you would like the Basic Residence Registration Card, you are supposed to ask at the counter of your local ward office after 8th July.
Please read the following PDF provided by the government for more detail.
Source (in English):