All U.S. green cards -- that is, the plastic identity card given to lawful permanent residents -- that were issued in 1989 or later have an expiration date on the front. They are valid for ten years, at the end of which you are expected to renew your green card. The process of doing so involves submitting Form I-90, Application to Replace Permanent Resident Card, issued by U.S. Citizenship and Immigration Services (USCIS).
This is a common source of confusion, but no, your status as a lawful permanent resident will not be affected if you do not renew your green card. However, you will face difficulty proving your status as a lawful permanent resident, getting employment, reentering the U.S. after foreign travel, and so forth. Besides, you are required by law to carry valid evidence of your immigration status -- and an expired green card is not considered valid evidence.
The renewal requirement comes as a particular surprise to immigrants who received green cards between 1979 and 1988. In the past, these did not have an expiration date and did not need to be renewed. However, the system has changed, and you are now expected to renew such cards. If you haven't done so already, do so right away.
NOTE: If your green card has a two-year expiration date on it, that's a completely different matter. You are not a permanent resident, but a "conditional resident," whose actual status will in fact expire if you do not take steps to convert to permanent residence within the 90 days before your expiration date. See an attorney if that date has gone by.
You should submit your application within the six months leading up to your green card's expiration. That will give USCIS time to process the application.
You must submit an application to renew your green card on USCIS Form I-90. (Don't be confused by the fact that this form is also used for other purposes, such as to request USCIS to replace errors on your green card, or to replace a lost card.) The form will ask you to provide such information as:
You must submit a copy of your expiring or expired green card along with Form I-90. If your name has changed since the issuance of your green card, you must submit evidence of your name change – court order, marriage certificate, adoption decree, and so on.
If you need help with the form, see How To Complete Form I-90 to Renew a Green Card
Submit the completed Form I-90 along with the required documents and a fee to USCIS; the address is on the I-90 page of its website. Alternatively, you can file the form online, but will still need to send your supporting documents to USCIS by mail.
You must also pay a biometrics fee for fingerprinting. Assuming your Form I-90 is complete, the next thing that will happen is that you will receive a letter from USCIS asking you to go to the local USCIS Application Support Center (ASC) for your biometrics appointment. Then they will check your record, to make sure you haven't committed any crimes or otherwise become deportable from the U.S. -- see Renewing Your Green Card After a Criminal Charge or Conviction for more information.
USCIS may contact you for more information or (in rare cases) call you for an interview before making a decision. It will notify you about its decision in writing.