If you have U.S. lawful permanent residence, you will have been issued a photo identity card proving your status, known as a "green card." That card has an expiration date. The time allotted before your card expires is ten years. After this time has passed, the green card holder is expected to renew the card (using Form I-90, issued by U.S. Citizenship and Immigration Services or USCIS).
But what if you are in the process of applying for U.S. citizenship when the green card expires?
A green card is merely evidence of your lawful immigration status. You are still a permanent resident, even if your green card has expired. On that basis, many people have chosen to simply apply for U.S. citizenship after their green card has expired, or to not renew the green card if it happens to expire while they're waiting for a USCIS decision on their citizenship application.
USCIS policy goes back and forth on how accepting it is of this practice, but lack of a renewed green card should not, by itself, hold up your naturalization approval.
Normally, you are expected to carry a valid green card with you at all times, as one of the responsibilities of being a permanent resident. Meanwhile, applying for citizenship or naturalization can be a months-long process. (The exact timeline depends on the USCIS office at which you apply.)
Spending all that time without a green card could make it difficult to gain employment in the U.S. (or prove your ongoing legal status to your current employer) and to reenter the United States if you travel abroad. Then again, the USCIS processing time to renew a green card also tends to be long; over a year is not uncommon.
If you really don't want to renew your green card, then you're on safest ground if the date that you apply for citizenship is more than six months prior to the date your green card will expire.
If concerned about any legal incidents that may cause your application for green card renewal or naturalization to put your legal status at risk, talk to an immigration lawyer before you proceed. Have an attorney review your recent history, legal permanent resident status, and immigration files to make sure you're not getting yourself into hot water with USCIS or the Department of Homeland Security.