A Zairyu Card is a Residence Card which is an ID card for non-Japanese residents. Foreign residents who are planning to live in Japan from medium to long-term stays receive a Zairyu Card upon arrival at Narita, Haneda, Chubu and Kansai airports.
Unless you apply for naturalization and gain Japanese citizenship, every foreigner residing in Japan must have a Zairyu Card.
You must keep this with you at all times and it can be used as your primary form of ID. Note that sometimes police officers can randomly stop you and ask to see your Zairyu Card. It will be a difficult situation if you are not carrying one when asked.
Having said that, there are some cases in which Residence Cards cannot be issued, including:
People with permission to reside in Japan for 3 months or less
People who are granted "Temporary Visitor" status
People who are granted "Diplomat" or "Official" status
People who are recognized by Ministry of Justice ordinance as equivalent to the foreign nationals in the aforementioned (1) to (3)
Special permanent residents
People who have no resident status