A green card is proof of a person's right to live and work in the US. If a person's green card is lost or stolen, it's a potentially troublesome situation. It is important to understand that loss of a green card does not mean they have lost their permanent resident status. However without the green card, problems are likely to arise while traveling or applying for jobs.
If you are in the United States when your green card is lost, it is important to fill out Form I-90, "Application to Replace Permanent Resident Card" as quickly as possible. It can take over six months to receive a new card.
If a person needs to travel to another country before their green card has been replaced, they will need to make an appointment with a USCIS office, and request a temporary green card stamp (I-551) for their home countries passport. A person should bring the Form I-797C, "Notice of Action" form that they received in reply to the I-90. The stamp is temporary proof of permanent status. The stamp is generally valid for one year. Employers are able to accept this stamp as proof of authorization to work.
If outside the United States when the green card is lost or stolen, one will need to notify a police department in the local jurisdiction to obtain a police report, and one will need to obtain carrier documentation, prepare Form I-131A, "Application for Travel Document (Carrier Documentation)", and file it at the nearest U.S. Embassy or Consulate to return to the United States.