Both U.S. citizens and lawful permanent residents who enter into marriage with a foreign-born person can help that person get a green card (lawful permanent residence). Among the basic requirements for a marriage-based green card is that the marriage be recognized as legally valid in the place where it occurred. A common law marriage, therefore, may be recognized for immigration purposes if it is legal in the country or U.S. state where the couple lived or currently live and led to the attainment of the same marital rights as are granted to traditionally married couples in that locale.
As background, the definition of a common law marriage is an agreement between a man and woman to become married without a formal civil or religious ceremony. When they take place in other countries, these are also sometimes referred to as "customary" or "tribal" marriages.
A minority of U.S. states allow common law marriages, for both opposite-sex and same-sex couples.
Contrary to popular myth, however, merely living together for a certain number of years in those states does not necessarily make you married. You must typically not only live together for a certain period of time, but intend to be married and hold yourselves out as a married couple in interactions with friends, relatives, neighbors, and the public, for example by using the same last name, introducing each other as "my husband" or "my wife," filling out official paperwork such as insurance documents, children's birth certificates, and joint tax returns in a way that indicates your status as a married couple, and so forth.
If your state offers you the opportunity to register your common law marriage with a local government agency, then the U.S. immigration authorities will expect you to have done so, and to provide proof of such registration.
In order to obtain a green card for an immigrant, you must not only show that the marriage is legally valid, but that it is bona fide—not a sham or fraud to get a green card. Like every married couple, you'll need to show multiple forms of documentary proof of your shared life.
Couples who have gone through a civil or religious ceremony commonly have photos of the ceremony, and show these to the immigration officials. Such photos certainly wouldn't be required in the case of a common-law marriage, and they're not the only type of proof the immigration authorities look for.
Other likely documents include joint lease or mortgage documents, joint bank or credit card statements, evidence of joint health, auto, or other insurance, affidavits from friends and relatives who've known you over the years, and so forth.
Nevertheless if, perhaps because you weren't sure whether your common law marriage was legally recognized you didn't take as many steps to combine these other aspects of your life, providing documentary proof of a common law marriage could be slightly more difficult than for traditionally married couples. You might also simply face uncertainty by immigration officials, or subjective opinions along the lines of, "If they really wanted their friends and family to know they were married, why didn't they just have a ceremony?"
Consider contacting an immigration lawyer to assess whether your common law marriage is likely to satisfy the U.S. immigration authorities and, if so, what sorts of documentary proof you'll need to gather to accompany the foreign-born spouse's application for a green card.