If you received your U.S. residency based on marriage to a U.S. citizen, but because of the recency of the marriage you were given conditional, rather than permanent residence (with a two-year expiration date), then you will need to fill out and submit USCIS Form I-751 to convert your conditional residency to permanent residency. See "Marriage-Based Conditional Residents: When and How to Apply for a Permanent Green Card" for details on the background of who needs to file and why.
Assuming you are eligible and ready to file the I-751 -- and you are still married and your U.S. spouse is cooperating with the process -- here is how to fill out the form. (If your spouse will not or cannot sign onto the form, see "What If Your U.S. Spouse Won't Sign the Joint Petition I-751?" for your next options.)
Part 1, “Information about you”: This refers to the immigrating spouse. The rest of this Part should be self-explanatory.
Part 2: If you are still married and the U.S. spouse is cooperating with the process of applying for your permanent residency, put an “X” in Box a.
Part 3: Mostly self-explanatory. You will probably know if you are in “removal or deportation proceedings.” You would be in this situation if you had been convicted of a crime, given USCIS reason to believe that your marriage is fraudulent, or done something else to make you deportable. If you are in court proceedings, you should already have a lawyer, who should help you with this petition as well.
If you check “yes” to the question about whether a fee was “paid to anyone other than an attorney in connection with this petition,” you have not done anything wrong, but that person needs to enter his or her name and other information.
Part 4: This Part refers to the petitioning spouse, not the immigrant. Self-explanatory.
Part 5: Self-explanatory.
Part 6: If you are requesting an accommodation at your interview because of a disability or impairment—such as having a sign-language interpreter—explain that here.
Part 7: It is very important that both of you remember to sign. Your two signatures are an indication of the ongoing validity of your marriage.
When you and your spouse are finished preparing the joint petition, make a complete copy for your files. Then send the packet of forms and documents to a USCIS Service Center—the address is in the instructions that come with the form, or you can call the USCIS information number at 800-375-5283.
As you may have learned by now, Service Centers are different than the local USCIS offices that you can actually visit. All contact with the Service Center will have to be by mail.
Using a courier service or certified mail with return receipt requested is highly recommended. USCIS Service Centers receive a huge volume of mail, and if your application gets lost, your proof of mailing will help convince USCIS to look for the petition or not penalize you for filing late.
Here are the items you will need to assemble for your joint petition to remove the conditions on your residence and become a U.S. permanent resident:
Required supporting documents, including:
A copy of your current green card (or I-551 in the language of USCIS); copy both the front and the back sides
Evidence of your marital relationship from the last two years (or slightly less, depending on when you're filing), such as copies of leases or mortgages in both your names, copies of joint bank, credit card, and utility statements or bills, copies of children's birth certificates, and so forth.
Application fee ($505 as of early 2015, but fees change often so double-check this at www.uscis.gov, or by calling 800-375-5283); you can mail a check or money order
Biometrics fee (for photos and fingerprints; $85 as of early 2015).