Historically, a "common law marriage" was a marriage that was created by agreement of the parties, often without even a formal ceremony, and not licensed by the government. That may not be a perfect definition, but it will be pretty close. Many (and perhaps all) states no longer recognize them. Nowadays, if two folks just "shack up" together, I don't know of any state that will treat them as married even if both had convinced themselves that they were "married" in the eyes of God, etc. Big government has created another monopoly for itself.
The Common Law is a much different thing. It is a description of the body of judge-made law that was developed over hundreds of years in Great Britain and subsequently in the United States. I described it more fully in the prior post.