What kind of baptism is essential to salvation, and what does 'essential' mean in this context? (Catholic perspective)

Baptism is necessary for salvation as Jesus himself said.

What is baptism? It is a valuntary acceptance of god. Catholic church believes, that its sacrament of baptism is the correct interpretation of these words.

To deny the sacrament, as a catholic, out of hubris is a sin and rejection of god.

To deny it, out of other reasons - god will judge.

What IS a catholic baptism? Just an acceptence of god. So what do you need for baptism according to catholic church?- desire for baptism. No oil, no water, no ritual no priest is required. However, when those are present, it is hubris to reject them.

The Nicene Creed professes one Baptism. This poses a challenge for people who believe in a Baptism of Water, Baptism of Blood and a Baptism of Desire. Frequently its supporters will say that one Baptism with three types. Baptism of Desire is a idea believed in by some saints who wondered what happened to a catechumen who died before Baptism. They wondered this specifically because the only options available were Hell or Limbo of the Children, since the Catholic Church took takes a literal view of John 3:5

Jesus answered, Amen, amen, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God.

Some saints like St. Alphonsus di Liguori believed in a Baptism of Desire for catechumens who explicitly desired it and were instructed in The Faith. However St. Gregory of Nazienzen when asked about this question mocked the questioner because he found the question absurd. I specifically say catechumens because there are certain things that you are required to believe to be saved, The Trinity and the Incarnation being among them.

Pope Eugene IV, Council of Florence, Athanasian Creed, Sess. 8, Nov. 22, 1439, ex cathedra: “Whoever wishes to be saved, needs above all to hold the Catholic faith; unless each one preserves this whole and inviolate, he will without a doubt perish in eternity.– But the Catholic faith is this, that we worship one God in the Trinity, and the Trinity in unity… “But it is necessary for eternal salvation that he faithfully believe also in the incarnation of our Lord Jesus Christ...the Son of God is God and man... This is the Catholic faith; unless each one believes this faithfully and firmly, he cannot be saved.”

So prior to Vatican II, there was never any mechanism where the pagan on the island could be saved, short of God sending an angel or bi-locating a saint to the island while the pagan was still alive and instructing him in the faith.

In modern times however the concept of Baptism of Desire, which is anathematized, has been used to kick open the door and the modern professing Catholics will say that a good Muslim or a good Hindu or a good Jew could get into Heaven, and the mechanism for this they will tell you is Baptism of Desire. They will frequently use an ambiguity in the Council of Trent to try to prove their point, however Trent anathematized their position.

Pope Paul III, The Council of Trent, Can. 2 on the Sacrament of Baptism, Sess. 7, 1547, ex cathedra: “If anyone shall say that real and natural water is not necessary for baptism, and on that account those words of Our Lord Jesus Christ: ‘Unless a man be born again of water and the Holy Spirit’ [John 3:5], are distorted into some sort of metaphor: let him be anathema.”

Now of course they will counter and say Desire isn't a metaphor, but I for one don't find that counter convincing.

Pope Paul III, The Council of Trent, Can. 5 on the Sacrament of Baptism, Sess. 7, 1547, ex cathedra: “If anyone says that baptism [the sacrament] is optional, that is, not necessary for salvation (cf. Jn. 3:5): let him be anathema.”

To answer your question directly water baptism is necessary for salvation.

Pope Pius IX, First Vatican Council, Sess. 3, Chap. 2 on  Revelation, 1870, ex cathedra: “Hence, also, that understanding of its sacred dogmas must be perpetually retained, which Holy Mother Church has once declared; and there must never  be a recession from that meaning under the specious name of a deeper understanding.”

My take on Catholic teaching is that it is conflicted on the issue, on the one hand wanting to take the stance that baptism is absolutely necessary for salvation, but also recognizing that there are real problems with that position.

The end result seems to be that it's necessary except where it's either practically impossible, or is prevented by circumstances outside of the believer's control. In other words, baptism is far and away the norm -- "I was going to get around to it one day", won't wash on judgement day.

Specifically, the CCC currently states that:

1257 The Lord himself affirms that Baptism is necessary for salvation. He also commands his disciples to proclaim the Gospel to all nations and to baptize them. Baptism is necessary for salvation for those to whom the Gospel has been proclaimed and who have had the possibility of asking for this sacrament. The Church does not know of any means other than Baptism that assures entry into eternal beatitude; this is why she takes care not to neglect the mission she has received from the Lord to see that all who can be baptized are “reborn of water and the Spirit.” God has bound salvation to the sacrament of Baptism, but he himself is not bound by his sacraments.

Holding baptism necessary for those "to whom the Gospel has been proclaimed and who have had the possibility of asking for this sacrament." This allows for other mechanisms in God's economy of which the church specifically decries knowledge.

1258 The Church has always held the firm conviction that those who suffer death for the sake of the faith without having received Baptism are baptized by their death for and with Christ. This Baptism of blood, like the desire for Baptism, brings about the fruits of Baptism without being a sacrament.

This then allows that the sacrament may be effected without the actual practice thereof, in special cases, where a greater sacrifice for Christ is made, in this case martyrdom.

1259 For catechumens who die before their Baptism, their explicit desire to receive it, together with repentance for their sins, and charity, assures them the salvation that they were not able to receive through the sacrament.

This conveys salvation via the genuine intent to be baptized, forestalled by unforeseen circumstance - the baptism of desire.

1260 “Since Christ died for all, and since all men are in fact called to one and the same destiny, which is divine, we must hold that the Holy Spirit offers to all the possibility of being made partakers, in a way known to God, of the Paschal mystery.” Every man who is ignorant of the Gospel of Christ and of his Church, but seeks the truth and does the will of God in accordance with his understanding of it, can be saved. It may be supposed that such persons would have desired Baptism explicitly if they had known its necessity.

This allows that those ignorant of the Gospel, yet obedient to it's precepts according to the revelation granted them may also see salvation.

1261 As regards children who have died without Baptism, the Church can only entrust them to the mercy of God, as she does in her funeral rites for them. Indeed, the great mercy of God who desires that all men should be saved, and Jesus’ tenderness toward children which caused him to say: “Let the children come to me, do not hinder them,” allow us to hope that there is a way of salvation for children who have died without Baptism. All the more urgent is the Church’s call not to prevent little children coming to Christ through the gift of holy Baptism

Lastly, this allows that those for whom baptism is physically impossible, the yet unborn children, the church allows for God to work salvation apart from baptism.

For myself, I note that Jesus himself made an exception for the thief crucified with him, which person I suppose falls under the provision of not having the possibility to be baptized before his death.

A note about infallible teachings (ex cathedra doctrine).

The doctrines on Baptism are not accorded this unique ex cathera status. See my answer to another question for more detail, including a link to a Vatican article explaining the ex cathedra exercise of the church's Magisterium. This other answer includes a longer list, for which I cannot find a Catholic source of authority (and it makes the erroneous claim that all Catholic doctrine is considered infallible).