If you are marrying a U.S. citizen, you will be eligible to apply for a green card after the marriage is official. What you must do to obtain a green card will depend on where you currently reside. Let's have a look at what this process entails.
If you're in the U.S.
If you already live in the U.S., you can apply for permanent residence after your marriage is official. You will apply on Form I-485, Application to Register Permanent Residence or Adjust Status. Simultaneously, your spouse will need to file Form I-130, Petition for Alien Relative. If you do this concurrently, you will be able to undergo the entire process in one step.
If you would rather apply in two steps, you can have your spouse file Form I-130 and wait until it is received or approved, at which time you will receive Form I-797 from USCIS. This will show your spouse's form has been received or approved. Then, you can file your own I-485 application with a copy of the I-130 receipt or the I-797 notice.
"Outside the U.S., you'll get a green card through consular processing."
If you're outside the U.S.
If you currently reside outside of the U.S., you'll need to get a green card through consular processing. This means USCIS and the Department of State will issue a visa on your I-130 petition when one becomes available. Then, you'll travel to the U.S. on that visa, at which time you'll become a permanent resident. After the Department of State receives your I-130, you will be notified that you are eligible to apply for a visa, which you must do within one year of that notification. Otherwise, you won't be eligible anymore.
In all cases
Filing to receive a green card through marriage will require you and your spouse to fill out the biographic information form G-325A. You'll also have to prove your marriage is in good faith (you may hear this referred to as "proving the bona fides of the marriage"). To prove this, you will need to submit copies of birth and marriage certificates, anything from the wedding you can use to prove it is genuine such as announcements and invitations, proof of jointly held assets and bank accounts and so on. Take the time to gather these documents together before you try to make your application so you are not delayed.