Certain categories of green card applicants must, after their initial visa petition is approved, wait for their priority date to become current before continuing on with their application for a green card. It's a fancy way of saying that you must wait until your number comes up on a waiting list, owing to the fact that numerous immigrants are in line ahead of you, in a green card category that's subject to annual limits. (Not everyone faces numerical limits or a wait -- for example, green card applicants in the immediate relative category receive unlimited numbers of visas annually.)
The people who will be given priority dates include those in the family-based or employment-based preference categories. The initial visa petition would have been filed by either their family member on Form I-130 ("Petition for Alien Relative"), or by an employer on Form I-140 ("Immigrant Petition for Alien Workers") following a successful application for labor certification. Certain highly qualified employees can self-file the Form I-140 petition.
When your priority date becomes current, you can either apply for adjustment of status if you are already in the United States in legal status or otherwise eligible to use the adjustment of status procedure, or apply for an immigrant visa at an U.S. Embassy or Consulate in your home country. Both lead to U.S. permanent residence (a green card), though in the latter case, you do not become a permanent resident until you actually use your immigrant visa to enter the United States.
The priority date is the date upon which either the U.S. Department of Labor received the labor certification application or U.S. Citizenship and Immigration Services (USCIS) received the completed Form I-130 visa petition or (in the case of self-petitioners) Form I-140.
If the visa petition or labor certification was incomplete when first filed, however, it will be returned to the petitioner and your priority date will be delayed.
If the visa petition for your green card is approved, USCIS will issue its approval on Form I-797, "Notice of Action." Your Form I-797 will contain your priority date.
To see what priority dates are becoming "current" -- meaning that people with those dates are now becoming eligible for visas/green cards -- you should check the Visa Bulletin published by the U.S. State Department every month, and available online. It shows a list of visa cutoff dates, in every category. In some cases, the categories are also broken down by country, due to the fact that there are per-country limits on visas, and the demand from those countries is especially high. This is usually true for China, India, Mexico, and the Philippines.
If the Visa Bulletin shows "C" or "current" under your application category and your country of citizenship, it means that there are visas available under the said category for your country and you should either expect to see correspondence from the National Visa Center (NVC) soon or should, if you're adjusting status, prepare and submit your green card application.
If the Visa Bulletin shows a date for your category and country, it means that all the people whose priority dates are prior to that date can apply for an immigrant visa under that category. So, when you see a date that's later than your own priority date, get ready to apply for your green card.
Consult with an experienced U.S. immigration attorney to learn more about your priority date and the timeline on which you're likely to be able to proceed with your green card application. The attorney can also help you with the application procedures for consular processing or adjustment of status and, if you're adjusting status, accompany you to your USCIS interview.