Tl;DR it's not really that different all things considered. It's just a term for old laws brought over during the establishment of the colonies and the subsequent state constitutions.

Common law in the US is the name for old English laws we established with the founding of the states. Many of our legal codes are based on common law Maryland and Pennsylvania are two common law states. That's not to say all the laws are outdated but you might hear old terms like "manslaughter" or "burglary" where newer legal codes make "x degree murder" equivalent to manslaughter and "breaking and entering" is equivalent to burglary

Though most often these days you'll hear it in terms of "common law marriage". This is a legal statute where if two single unmarried persons cohabitate for a period of time, then the possessions and dwelling can be considered the property of both people equally just like marriage. The "common law" part simply refers to it being a carry over from the codes established using older British codes.