Common laws are those laws which are based on precedent. Civil laws are those laws which apply to civil actions or charges that someone has had their civil rights violated.
For example, common law marriage is a law which allows for couples to be considered married after living together for a certain number of years. While this is not the common law in many jurisdictions, it does exist in some states.
Common laws are those which are based on precedent, rather than actual statutes. In common law marriages, for example, the couple has lived as husband and wife and is therefore considered to be, even though they have not legally been married.
Civil laws are those laws which protect civil rights. For example, some couples claim that the inability for same sex couples to marry is a violation of their civil rights. In some jurisdictions, same sex couples may be involved in a common law marriage, but this is still a violation of their civil rights when they wish to be legally married.