If you are a J-1 visa holder planning to marry a U.S. green card holder (lawful permanent resident), you yourself might eventually be eligible for a family-based U.S. green card. However, it's highly likely you will need to spend at least a few years outside the United States first, particularly if your J-1 stay has run out, for the following three reasons:
Fortunately, you don't have to add up all the time delays described above into one grand total, but can wait them out simultaneously.
As the spouse of a U.S. green card holder, you are eligible for a green card, but not until a visa number becomes available. In the meantime, you will be placed on a waiting list, based on your "priority date." That's the date your green-card-holding spouse filed the required initial visa petition with U.S. Citizenship and Immigration Services (USCIS) on Form I-130.
The wait varies depending on demand, typically ranging from between zero and five years. The only thing you can do to speed it up is if your spouse successfully applies for naturalization and becomes a U.S. citizen. (See When Can I Apply for U.S. Citizenship?.) In that case, you would become an "immediate relative," and could apply for your green card without further delay.
For more information, see Applying for a Green Card Based on Marriage to a U.S. Lawful Permanent Resident.
The J-1 visa was created in order to foster international exchanges of knowledge. For that reason, you might be expected to return home and share that knowledge once your visa expires; particularly if your home country helped fund your U.S. stay. There's a good chance that your J-1 visa came with an automatic requirement that you return home, and stay there, for two years following your U.S. stay before attempting to return to the United States.
Although waivers are available, you could only make use of one if you had an immediate right to return, such as a marriage to a U.S. citizen rather than a green card holder.
Due to many legal consequences, it is never advisable to remain in the United States beyond the permitted time on a visa. Once your permitted stay expires, your visa is automatically cancelled and you are in the U.S. unlawfully. Many people who overstay feel like they are getting away with something when no one from the immigration department shows up to deport them, not realizing that the consequences usually come later.
For example, if you stay beyond your expected departure date by six months, and then leave the U.S. (as you'll most likely have to do), you will not be allowed to return to the U.S. for three years, despite your marriage. If your overstay lasts a year or more, you will not be allowed to return for ten years.
Immigration law is always complicated, and it is wise to hire an immigration attorney for situations such as this one. The attorney will work to analyze the best strategy for you and your spouse, and help you to complete all the required paperwork.