The current style of U.S. green card (also known as an I-551 or permanent resident card) expires every ten years, at which time it must be renewed. The previous style of green card did not contain an expiration date -- but is no longer considered valid, so it too needs to be renewed.
If you did not renew your green card on time, you should do so as soon as possible. Don't worry that you have lost your permanent residence -- the green card is merely evidence of your status, and letting it expire does not damage that underlying status.
The key thing to understand, however, is that one of your obligations under U.S. immigration laws is to carry a valid green card with you at all times. If you are caught with an expired green card, or an old-style green card with no expiration date, you could be prosecuted for a misdemeanor. Having a crime on your record can lead to other problems, for example when you apply for U.S. citizenship.
To renew your green card, you will need to fill out and submit Form I-90, issued by U.S. Citizenship and Immigration Services. You can submit it either online or by mail. After submitting the application, you will be called in to have your fingerprints taken, in order to check your record for any crimes or immigration violations.
See How To Complete Form I-90 to Renew a Green Card for tips on the paperwork.
If you have committed any crime or done anything else to violate your legal right to remain in the country, you may have become removable (deportable). Typically felony crimes cause problems, but a misdemeanor can raise red flags as well. The fingerprint check will reveal crimes or immigration violations that took place since you last received your green card, and applying to renew could result in your being placed in removal proceedings. For more information, see Renewing Your Green Card After a Criminal Charge or Conviction. You'd also be wise to consult an immigration lawyer.
If you are already eligible for U.S. citizenship, you might want to apply for naturalization without renewing your green card first -- a practice that USCIS is not happy about, but has historically allowed. Once you obtain citizenship, you no longer have to renew your green card or worry about it expiring.
See Applying for U.S. Citizenship With an Expired Green Card for a discussion on that option.
If you are concerned about any legal incidents that may cause your application for green card renewal to put your legal status at risk, it is a good idea to hire an immigration lawyer. The attorney can help you confirm your permanent resident status, assess whether any risks present themselves in your case, and prepare the application for renewal.