The Mosaic Code or Law of Moses is a denomination used to refer to the Torah, the holy book of Jewish people. It is believed that it was written by Moses, leader of the Israelites during their exodus from Egypt.
Hammurabi's Code is a Babylonian code of laws from 1754 BCE. It comprises 282 laws, that include scaled punishments, adjusting the lemma "an eye for an eye, a tooth for a tooth" depending on the social status of each person, slave or free, man or woman, etc. It includes sections that deal with contract law, family law, etc.
Juris Civilis is a recopilation of jurisprudence legal texts, enacted between from 529 and 534 under the leadership of Emperor Justinian I, who governed the Eastern Roman Empire. It is also known as the Code of Justinian.
The Common Law received this denomination because it was shared ("common") to all the king's courts across England, in the centuries after the Norman Conquest that took place in 1066. Subsequently, the British Empire spread this legal system to its colonies, and many still have a system based on the original common law nowadays, as it is the case of the US.