The Law of Moses is the list of rituals, moral laws, and guidance that are recorded in the the Books of Exodus, Leviticus, Numbers, and Deuteronomy. The are called the Law of Moses because God gave them to the nation of Israel through a man called Moses:
Remember the law of Moses My servant, even the statutes and ordinances which I [God] commanded him in Horeb for all Israel. (Mal. 4:4, NASB)
The Jews were to follow the Law of Moses (Deut. 4:1).
The New Testament tells us that God added (Gal. 3:17,19) the Law of Moses to the life of his people to teach them: the Law taught people about sin (Gal. 3:19, 22-23), and it taught people about Christ (Gal. 3:24, NASB). Now that Christ has come, the Law has been abolished (Gal. 3:25; Rom. 10:4: i.e. we do not need to follow the Law of Moses nowadays). Instead, nowadays we need to follow the Lord Jesus Christ himself (Gal. 6:2; Phil. 2:5; 1Peter 2:21).