Ezekiel 36:25
[Prophecy] Then will I sprinkle clean water upon you, and ye shall be clean: from all your filthiness, and from all your idols, will I cleanse you.

Acts 2:38
Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost.

Acts 22:16
And now why tarriest thou? arise, and be baptized, and wash away thy sins, calling on the name of the Lord.

Like all Sacraments (the other 6 being Eucharist, Confession, Confirmation, Holy Orders, Matrimony, and Unction), Baptism is not a work of man; it is a work of Christ, an act of His grace:

Titus 3:5-7
Not by works of righteousness which we have done, but according to his mercy he saved us, by the washing of regeneration, and renewing of the Holy Ghost; Which he shed on us abundantly through Jesus Christ our Saviour; That being justified by his grace, we should be made heirs according to the hope of eternal life.

It is through Baptism that we are "born again" (or "born from above") of "water and of the Spirit" (John 3:3-5). Read the entire chapter of John 3 which speaks of being "born again" and please note that it is all about Baptism. Despite what some Protestants believe, being "born again" doesn't mean "having an emotional high" or "making a decision for Christ," though these are fine and good, the latter being necessary after the age of reason; being "born again" very clearly refers to Baptism of water and of the Spirit. This regeneration of water and Spirit is necessary to enter the Kingdom of God:

John 3:3-5
Jesus answered and said unto him, Verily, verily, I say unto thee, Except a man be born again, he cannot see the kingdom of God. Nicodemus saith unto him, How can a man be born when he is old? can he enter the second time into his mother's womb, and be born? Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God

Why is water necessary? Why would Almighty God require us to use such a wordly element? First, His creation, though now fallen, is good, and the Holy Spirit sanctifies that which is fallen. We are not to approach nature with a dualist, gnostic mindset that sees His creation as "utterly corrupt" and inherently evil, and we cannot deny the power of God to use mere things for our good. Second, ultimately, it's not for us to question why, though it is fascinating to ponder; it's ours to do what He tells us. St. John Chrysostom (A.D. 347-407) answers that:

If any enquire, "Why is water included?" let us also in return ask, "Wherefore was earth employed at the beginning in the creation of man?" for that it was possible for God to make man without earth, is quite plain to every one. Be not then over-curious. That the need of water is absolute and indispensable, you may learn in this way. On one occasion, when the Spirit had flown down before the water was applied, the Apostle did not stay at this point, but, as though the water were necessary and not superfluous, observe what he says; "Can any man forbid water, that these should not be baptized, which have received the Holy Ghost as well as we?" (Acts 10:47)

The Old Covenant was entered into through circumcision; the New Covenant is entered into through Baptism:

Colossians 2:11-12
In whom also ye are circumcised with the circumcision made without hands, in putting off the body of the sins of the flesh by the circumcision of Christ: Buried with him in baptism, wherein also ye are risen with him through the faith of the operation of God, who hath raised him from the dead.

St. John Chrysostom (A.D. 347-407) writes this of the above verses:

See how near he is come to the thing. He saith, "In the putting" quite away, not putting off merely. "The body of sins." He means, "the old life." He is continually adverting to this in different ways, as he said above, "Who delivered us out of the power of darkness, and reconciled us who were alienated," that we should be "holy and without blemish." (Col. i. 13, 21.) No longer, he saith, is the circumcision with the knife, but in Christ Himself; for no hand imparts this circumcision, as is the case there, but the Spirit. It circumciseth not a part, but the whole man. It is the body both in the one and the other case, but in the one it is carnally, in the other it is spiritually circumcised; but not as the Jews, for ye have not put off flesh, but sins. When and where? In Baptism.

Just as children were once circumsized as infants, they are now baptized as infants because the Kingdom of God, which is entered into through Baptism, most certainly includes them:

Luke 18:16-17
But Jesus called them unto him, and said, Suffer little children to come unto me, and forbid them not: for of such is the kingdom of God. Verily I say unto you, Whosoever shall not receive the kingdom of God as a little child shall in no wise enter therein.

Acts 2:38-39
Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost. For the promise is unto you, and to your children, and to all that are afar off, even as many as the LORD our God shall call.

So in the New Testament, entire households were baptized:

Acts 16:14-15
And a certain woman named Lydia, a seller of purple, of the city of Thyatira, which worshipped God, heard us: whose heart the Lord opened, that she attended unto the things which were spoken of Paul. And when she was baptized, and her household, she besought us, saying, If ye have judged me to be faithful to the Lord, come into my house, and abide there. And she constrained us.

Acts 16:25-34
And at midnight Paul and Silas prayed, and sang praises unto God: and the prisoners heard them. And suddenly there was a great earthquake, so that the foundations of the prison were shaken: and immediately all the doors were opened, and every one's bands were loosed. And the keeper of the prison awaking out of his sleep, and seeing the prison doors open, he drew out his sword, and would have killed himself, supposing that the prisoners had been fled. But Paul cried with a loud voice, saying, Do thyself no harm: for we are all here. Then he called for a light, and sprang in, and came trembling, and fell down before Paul and Silas, And brought them out, and said, Sirs, what must I do to be saved? And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house. And they spake unto him the word of the Lord, and to all that were in his house. And he took them the same hour of the night, and washed their stripes; and was baptized, he and all his, straightway. And when he had brought them into his house, he set meat before them, and rejoiced, believing in God with all his house.

And the early post-New Testament Church carried on. There was no question as to whether or not infants should be baptized, though there was debate among a few as to whether they should wait to be baptized on the 8th day (the day children were circumcised in the Old Covenant), a concept the Church rejected. This is what St. Cyprian of Carthage (baptized ca. A.D. 246) wrote on the topic:

But in respect of the case of the infants, which you say ought not to be baptized within the second or third day after their birth, and that the law of ancient circumcision should be regarded, so that you think that one who is just born should not be baptized and sanctified within the eighth day, we all thought very differently in our council. For in this course which you thought was to be taken, no one agreed; but we all rather judge that the mercy and grace of God is not to be refused to any one born of man. For as the Lord says in His Gospel, "The Son of man is not come to destroy men's lives, but to save them," as far as we Can, We must strive that, if possible, no soul be lost. For what is wanting to him who has once been formed in the womb by the hand of God? To us, indeed, and to our eyes, according to the worldly course of days, they who are born appear to receive an increase. But whatever things are made by God, are completed by the majesty and work of God their Maker.

Moreover, belief in divine Scripture declares to us, that among all, whether infants or those who are older, there is the same equality of the divine gift. Elisha, beseeching God, so laid himself upon the infant son of the widow, who was lying dead, that his head was applied to his head, and his face to his face, and the limbs of Elisha were spread over and joined to each of the limbs of the child, and his feet to his feet. If this thing be considered with respect to the inequality of our birth and our body, an infant could not be made equal with a person grown up and mature, nor could its little limbs fit and be equal to the larger limbs of a man. But in that is expressed the divine and spiritual equality, that all men are like and equal, since they have once been made by Godl; and our age may have a difference in the increase of our bodies, according to the world, but not according to God; unless that very grace also which is given to the baptized is given either less or more, according to the age of the receivers, whereas the Holy Spirit is not given with measure, but by the love and mercy of the Father alike to all. For God, as He does not accept the person, so does not accept the age; since He shows Himself Father to all with well-weighed equality for the attainment of heavenly grace.

Baptism leaves an indelible mark on the soul, so one may be baptized only once.

Ephesians 4:4-6
There is one body, and one Spirit, even as ye are called in one hope of your calling; One Lord, one faith, one baptism, One God and Father of all, who is above all, and through all, and in you all.

The ordinary minister of Baptism is the Bishop (and, secondarily, the priest), and adults are usually baptized at Easter time in the context of the beautiful Rite of Baptism, and after a period of catechesis. In emergencies, though, a person may be baptized anytime, and by anyone (the efficacy of the Sacraments does not depend on the personal holiness of the minister) who:

• intends to do what the Church does,
  
  
• uses the proper matter (water), which is poured such that it touches at least the forehead of the initiate, and who, while pouring, sprinkling, or immersing,
  
  
• says the words "I baptize thee in the Name of the Father, and of the Son, and of the Holy Ghost." ¹

Matthew 28:19
[Jesus speaking] Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost.

Therefore, if one is baptized as described above, that Baptism is valid and need not be repeated in order for one to become fully Catholic. If one is not sure that one is baptized, if there is any uncertainty at all that the proper form and matter were not used, one is baptized conditionally with water and the words: "If you are not baptized, I baptize thee in the Name of the Father, and of the Son, and of the Holy Ghost."

If Water Baptism is Impossible

Those who vow to receive Baptism but are never in a position to receive it (e.g., a man stranded on a desert island) may, by the mercy of Christ, be given the fruits of Baptism. This doesn't obviate obedience to the command to the Apostles if and when it is possible to obey, but shows clearly that, though we are bound by the Sacraments, God is not. We call this "Baptism of Desire," which is, according to the Catholic Encyclopdia, "a perfect contrition of heart, and every act of perfect charity or pure love of God which contains, at least implicitly, a desire (votum) of baptism."

There is also the Baptism by Blood, which St. Augustine (A.D. 354-430) describes like this:

When any die for the confession of Christ without having received the washing of regeneration, it avails as much for the remission of their sins as if they had been washed in the sacred font of baptism.

Immersion only?

Some Protestants argue that "baptizo" in the New Testament means "immersion" and that any Baptism that doesn't include immersion is not a true Baptism. While "baptizo" does mean "immersion," it also means "washing," as is evident in this verse:

Luke 11 38
And when the Pharisee saw it, he marvelled that he had not first washed [baptizo] before dinner.

Trust me, first century Jews didn't immerse themselves before dinner, and Ezekiel's prophecy mentioned above also not only includes, but specifically mentions "sprinkling":

Ezekiel 36:25
[Prophecy] Then will I sprinkle clean water upon you, and ye shall be clean: from all your filthiness, and from all your idols, will I cleanse you.

The verses mentioned above that describe St. Paul's baptizing people in their households -- houses without swimming pools in them -- indicates "washing" rather than "immersion." Paul himself was baptized not only in a house, but standing up:

Acts 9:17–18
And Ananias went his way, and entered into the house; and putting his hands on him said, Brother Saul, the Lord, even Jesus, that appeared unto thee in the way as thou camest, hath sent me, that thou mightest receive thy sight, and be filled with the Holy Ghost. And immediately there fell from his eyes as it had been scales: and he received sight forthwith, and arose, and was baptized.

Baptism by immersion was a very common practice in the early Church -- the most common practice, in fact; but it wasn't the exclusive practice, as the Bible attests. Most Catholic churches had baptisteries in which the catechumen would stand and either be immersed (if the size of the baptistery allowed) or have water poured over his head, but all three methods -- immersion, pouring, or sprinkling -- were used. The earliest extra-Biblical writing we have on the topic is the Didache, a 1st c. document known as "The Teachings of the Twelve Apostles." On the topic of Baptism, it reads:

But concerning baptism, thus baptize ye: having first recited all these precepts [i.e., all that is included in the Rite of Baptism], baptize in the name of the Father, and of the Son, and of the Holy Spirit, in running water; but if thou hast not running water, baptize in some other water, and if thou canst not baptize in cold, in warm water; but if thou hast neither, pour water three times on the head, in the name of the Father, and of the Son, and of the Holy Spirit. But before the baptism, let him who baptizeth and him who is baptized fast previously, and any others who may be able. And thou shalt command him who is baptized to fast one or two days before.

Please, if you believe in the Lord Jesus Christ but have not been baptized, be baptized as soon as possible, and have your children baptized, too!

Further Reading

The Sacrament of Baptism
Infant Baptism
Born Again
Triune Baptism