For the law was given through Moses; grace and truth came through Jesus Christ.
“the law” strictly speaking was the Pentateuch, the first 5 books of the Old Testament. It came to mean the whole of scripture and thus, the whole of Judaism. Here it probably takes on the last meaning; John is contrasting Judaism with Christianity. Both came from God and both came through an agent, but note the different verbs used: the law was given through Moses, but grace and truth came through Jesus Christ. The verbs associate grace and truth more closely with Jesus than the Law with Moses. Compare also the content of Judaism with that of Christianity: Judaism is represented as the law; Christianity as grace and truth.
The use of the double name “Jesus Christ” accomplishes 2 things: it points out that the Christ (Messiah, or anointed One) of God is Jesus; and it states that Jesus is also the Word become flesh. Although this is the first time John uses the name Jesus, it is certainly not the last: he uses it 237 times in his gospel. The purpose John writes this gospel is so that men might believe that Jesus is the Christ of God.
John 20:31 But these are written that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in his name.