go and (literally going): This is a command to make disciples. Matthew likely meant "go" as a separate command, not just as "while you go"(compare similar uses in Matthew 9:6; Acts 10:20).
make disciples: A disciple turns from sin, trusts Jesus for salvation, and obeys his teachings.
all nations: The gospel mission to non-Jewish (Gentile) people officially started here (see Matthew 10:5–6; 15:24; also 1:1; 2:1–12; 8:5–13; 21:43; 24:14; 26:13).
Two parts of making disciples are baptizing and teaching (Matthew 28:20). Baptism is a public identification with Jesus and submission to him and his work of salvation (see 3:13–17; 10:32–33; 20:28; 26:28; Acts 2:38; Romans 6:1–11; 1 Peter 3:21).
in the name of: This means entering a relationship and its benefits for true believers in Christ.
Father … Son … Holy Spirit: This phrase shows the Christian belief that the one God is three persons (Father, Son, and Holy Spirit) known as the Trinity (see also 1 Corinthians 12:4–6; 2 Corinthians 13:14; Ephesians 4:4–6; 2 Thessalonians 2:13–14; 1 Peter 1:2; Revelation 1:4–6).