Another potential essay question: What Doctrine of Baptism emerges from the Fathers of the first two centuries.
The Fathers of the Church in the first two centuries had expounded the Trinitarian formula found in Matthew's Gospel, Chapter 28: Go, therefore, and make disciples of all nations, baptizing them in the name of the Father, and of the Son, and of the holy Spirit, teaching them to observe all that I have commanded you.
Thus, if you are coming into full communion with the Catholic Church, it is not necessary to be "re-baptized" if you already had a legitimate baptism following the Trinitarian formula. Most Protestant communities baptize legitimately and therefore those coming in would not need to be re-baptized. It's not theologically possible. Baptism can only be done once. Even though Protestants have a different understanding of baptism, the baptism itself is valid. However, if a person is coming from a non-Trinitarian community such as Mormonism, the person has never been legitimately baptized and would require it.
In the first century document, the Didache (A.D. 80)re-affirms the Trinitarian formula. It also suggests that the baptism be in living water (Jesus was baptized in the Jordan River--I renewed my baptismal vows there this past May!), if there is no living water, then in other water, if neither are available it instructs to "pour water three times on the head" using the Trinitarian formula. It also exhorts the baptized to fast for one or two days. Thus, the sacrament is instituted by Christ and is a real expression of entering into the life of the Triune God in which water is essential. The exhortation to fast is more a discipinary, rather than doctrinal matter. I do not believe it is too much emphasized these days perhaps because most baptisms in the Church are infant baptisms.
In the second century, St. Ignatius of Antioch writes to the Smyrnaeans that baptism can not take place without the permission of the Bishop. Today, I am not sure how that works. But again, I do not think that is a doctrinal issue.
Also in the second century, the Letter of Barnabas beautifully expresses, "In this way He says that we descent into the water full of sins and foulness, and we come up bearing fruit in our heart, having fear and hope in Jesus in the Spirit."
Doctrinally speaking, this seems to imply that there is a real change that accompanies baptism. It is not just an external expression of inward faith (it is that too), but a cleansing and remission of sin and a receiving of grace.
This real change is also expressed by St. Justin Martyr of the second century, "Their washing is called illumination, because it enlightens the intelligence of those who learn these things."
Lastly, this beautiful treatise on baptism of Tertullian's of the second/third century must be not be kept hidden: "The sins of our earlier blindness are washed away and we are released for eternal life will not be superfluous...Vipers and asps, as it is true of serpents in general, are found in dry and waterless places. But we little fishes, are born in water after the manner of our (insert the Greek word for FISH, which was the early Church's symbol for the Faith), Jesus Christ; nor can we be otherwise saved, except by abiding permanently in the water."
Tertullian's beautiful words imply that something supernatural does in fact happen during baptism and that it is necessary for salvation. (Of course, there is baptism of blood, desire, etc. but that is another subject entirely and they don't count if baptism by water/spirit is available...no lingering allowed; but I digress.)
A couple of issues regarding baptism in the first two centuries (and following) that have since been clarified by the Church regard the delaying of baptism (Tertullian discusses this)and the one time remission of sins in baptism (Hermas discusses this). Both are related. Since the early Church believed that remission of sins could only come once (or twice, sometimes personal confession was allowed one time after baptism)with baptism, it became a practice to delay baptism until adulthood or until "you got sin out of your system," in a sense. This is why Constatine delayed baptism and even St. Augustine in the fifth century.