As far as I remember, there is no rule in the Bible as to who can baptize a new believer. However, I'm almost certain that sometime in the early history of the Roman Catholic church, someone or some group of people decided on some rule, such as being a pastor. Or, perhaps they instead decided that anyone could baptize a new believer. I don't know, so I'm asking: according to the Roman Catholic church, who can baptize a new believer?
In an emergency where there is danger of imminent death, anyone (yes, anyone) who has the appropriate intention can baptize, by using the Trinitarian baptismal formula.
In all other cases, Catholics are instructed to contact their parish.
(Note that this only concerns who can baptize: not all persons should be baptized. There are certain restrictions regarding the baptism of certain adults, though young children can always be baptized.)
The official Catechism addresses this directly:
1256 The ordinary ministers of Baptism are the bishop and priest and, in the Latin Church, also the deacon. In case of necessity, anyone, even a non-baptized person, with the required intention, can baptize, by using the Trinitarian baptismal formula. The intention required is to will to do what the Church does when she baptizes. The Church finds the reason for this possibility in the universal saving will of God and the necessity of Baptism for salvation.
To be very clear: in cases where there is danger of imminent death, anyone can baptize. A person does not need to be male or baptized. Even a child can baptize in an emergency, if an adult is not present or incapacitated, though the child must be able to form the appropriate intention.
When there is no danger of imminent death, baptism is to be handled by the Church. The relevant Canon law is:
THE MINISTER OF BAPTISM
Can. 861 §1. The ordinary minister of baptism is a bishop, a presbyter, or a deacon, without prejudice to the prescript of can. 530, n. 1.
Can. 530 The following functions are especially entrusted to a pastor:
1/ the administration of baptism; [...]
§2. When an ordinary minister is absent or impeded, a catechist or another person designated for this function by the local ordinary, or in a case of necessity any person with the right intention, confers baptism licitly. Pastors of souls, especially the pastor of a parish, are to be concerned that the Christian faithful are taught the correct way to baptize.
Can. 862 Except in a case of necessity, no one is permitted to confer baptism in the territory of another without the required permission, not even upon his own subjects.
Can. 863 The baptism of adults, at least of those who have completed their fourteenth year, is to be deferred to the diocesan bishop so that he himself administers it if he has judged it Expedient.
These rules are in place to ensure that the baptism is, above all, performed in a valid way, but also to ensure that it is recorded appropriately.
It would be unwise and prideful for a person to perform baptism out of accordance with these rules, for the same reason that it would be unwise and prideful for a person to perform life-threatening surgery without the proper consultation of a trained doctor. Though of course, when a person is dying in front of you and there is nobody better able, you will, can, and are obliged to do everything you can to help save them, if they can be saved.
There is probably a better source, but from my Ad Altare Dei handbook* it says that any baptized male in the Catholic church can baptize another person into the Catholic Church, however, when available an ordained priest is preferred.
*Link is to a description of the program, it is a program for the Catholic Scouting religious award covering the Sacraments. The handbook is an official publication from the Roman Catholic Church and distributed from the scout store or your local Diocese.
There is another aspect to this question which is important. It is that the Catholic Church accepts as valid any baptism done in the Trinitarian form with the proper intention. This means that if an adult from another Christian denomination who has already been baptized converts to the Catholic Church, in most cases, he or she does not need to be rebaptised (or even conditionally rebaptised). Thus, in this sense, there is no such thing as a Catholic baptism per say but just valid Christian baptism.
Therefore, the answer is "anyone who follows the proper Trinitarian form, with the proper intention.