The best place to go for the Lutheran view of baptism is Martin Luther himself. His Small Catechism gives a brief yet profound explanation:
What is Baptism?
Baptism is not just plain water, but it is the water included in God's command and combined with God's word.
Which is that word of God?
Christ our Lord says in the last chapter of Matthew: "Therefore, go and make disciples of all nations baptizing them in the name of the Father, and of the Son, and of the Holy Spirit."
What benefits does Baptism give?
It works forgiveness of sins, rescues from death and the devil, and gives eternal salvation to all who believe this, as the words and promises of God declare.
Which are these words and promises of God?
Christ our Lord says in the last chapter of Mark: "Whoever believes and is baptized will be saved, but whoever does not believe will be condemned."
How can water do such great things?
Certainly not just water, but the word of God in and with the water does these things, along with the faith which trusts this word of God in the water. For without God's word the water is plain water and no Baptism. But with the word of God it is a Baptism, that is, a life giving water, rich in grace, and a washing of the new birth in the Holy Spirit, as St. Paul says in Titus chapter three:
"He saved us through the washing of rebirth and renewal by the Holy Spirit, whom He poured out on us generously through Jesus Christ our Savior, so that, having been justified by His grace, we might become heirs having the hope of eternal life. This is a trustworthy saying." (Titus 3:5-8)
What does such baptizing with water indicate?
It indicates that the Old Adam in us should by daily contrition and repentance be drowned and die with all sins and evil desires, and that a new man should daily emerge and arise to live before God in righteousness and purity forever.
Where is this written?
St. Paul writes in Romans chapter six: "We were therefore buried with Him through baptism into death in order that, just as Christ was raised from the dead through the glory of the Father, we too may live a new life." (Romans 6:4)
What we believe about baptism:
Baptism is essentially a means by which He has chosen to bring us His Spirit and the forgiveness of sins. God often uses things which seem ordinary to do miraculous works. He speaks to us through a book. He came to us in human flesh. He even spoke through an ass! God often hides Himself in ordinary elements as He reveals Himself. This is the same with the water of baptism.
We believe in baptismal regeneration. This means that the Spirit has chosen to work through baptismal water in the same way that He works through His word. Reformed Christians often say that the preached word is a means of regeneration but baptism is not. We believe that both are means which God uses to bring His promise to us. Baptism is the gospel in visible form, thus it gives all of the benefits of the gospel.
We believe in infant baptism. Since infants cannot understand the word, God uses baptism as a means to regenerate them and bring them into the faith. Through it, God gives faith. If faith is truly a gift of God and not a human work, God can certainly do this for an infant. He can also do it through whatever means He has chosen.
We believe that baptism is a form of the gospel, not a form of the law. Baptism is an act performed by Christ, through the hands of the administer of the sacrament. It is His gift of life and salvation. It is not a work we do. It is not something we do to profess our faith, or to profess that we will raise our children in the faith. It is a gift of grace through the promise of the gospel.
What we do not believe:
We do not believe that baptism is absolutely necessary for salvation. Since God works through both word and sacrament, the word is sufficient to regenerate and save. However, if one refuses to get baptized, this is evidence that he was never saved since he is denying what Christ has commanded.
We do not believe the Roman Catholic view of baptism. The Roman Catholic church denies that faith is necessarily given at baptism. They also deny that sin remains after baptism.
We do not believe that everyone who was ever baptized will be saved. If one rejects God's offer through baptism, or does not continue in the faith given at baptism, his baptism becomes a means of judgement rather than salvation.
This does not mean that we deny justification by faith alone because we believe baptism saves. The issue is that baptism and faith are not separate things. Baptism gives and strengthens faith. Baptism also delivers the promise which faith clings to.
These are the main points of the Lutheran view of baptism and how it differs from both the Reformed and Roman Catholic teachings on the subject.