Most Catholics are baptized as infants; consequently, there are two sets of requirements: one for the parents or guardians of infants who are to be baptized, and one for adults (those who have "reached the age of reason", which is not specified in the Code of Canon Law and may be perhaps left up to the local bishop).
The general church-wide rules regarding baptism are found in Canons 850 to 878 of the Code of Canon Law
The Church baptizes only once, so that only those who have never been baptized are eligible for baptism (cf. Canon 864). Typically, those who have been baptized in a Protestant or Orthodox denomination are considered by the bishop to have been baptized.
With respect to infants:
- Since a name is typically bestowed at baptism, the priest must make sure that it's not a completely anti-Christian name (no swear words as names, no naming your child after Satan, etc. (cf. Canon 855)
- At least one of the parents or guardians must consent
- It must be at least reasonably likely (in the opinion of the priest) that the child will be brought up Catholic
- The parents are to attend some sacramental preparation classes (the exact requirements for what is taught there, how long, and so on are up to the local bishop)
(Note: if either of these last two are not fulfilled, the baptism is valid; that is, the person is still considered as baptized. However, it is illicit; it's not in the proper format. Illicit baptisms could be performed in emergencies, say, in danger of death; but they are not supposed to be the norm.)
With respect to adults:
- Generally speaking, they must have some reasonable amount of instruction in the faith. Canon 865, section 1, states:
For an adult to be baptized, the person must have manifested the intention to receive baptism, have been instructed sufficiently about the truths of the faith and Christian obligations, and have been tested in the Christian life through the catechumenate. The adult is also to be urged to have sorrow for personal sins.
There are also requirements for sponsors (the Catholic church requires them; for infants there are usually two, who must be a man and a woman, and they are typically known as godparents; for adults there is usually only one), and requirements for who may do the baptism. But these are the specific Church-wide requirements, so far as they go, for the person baptized.
Here, as requested in comment, are the specific requirements word-for-word out of the Code of Canon Law:
Book IV: Function of the Church
Part I: The Sacraments
Title I: BAPTISM
THOSE TO BE BAPTIZED
Canon 864 Every person not yet baptized and only such a person is capable of baptism [i.e. is eligible to be baptized].
Canon 865 §1. For an adult to be baptized, the person must have manifested the intention to receive baptism, have been instructed sufficiently about the truths of the faith and Christian obligations, and have been tested in the Christian life through the catechumenate. The adult is also to be urged to have sorrow for personal sins.
§2. An adult in danger of death can be baptized if, having some knowledge of the principal truths of the faith, the person has manifested in any way at all the intention to receive baptism and promises to observe the commandments of the Christian religion.
Canon 866 Unless there is a grave reason to the contrary, an adult who is baptized is to be confirmed immediately after baptism and is to participate in the eucharistic celebration also by receiving communion.
Canon 867 §1. Parents are obliged to take care that infants are baptized in the first few weeks; as soon as possible after the birth or even before it, they are to go to the pastor to request the sacrament for their child and to be prepared properly for it.
§2. An infant in danger of death is to be baptized without delay.
Canon 868 §1. For an infant to be baptized licitly:
1. the parents or at least one of them or the person who legitimately takes their place must consent;
2. there must be a founded hope that the infant will be brought up in the Catholic religion; if such hope is altogether lacking, the baptism is to be delayed according to the prescripts of particular law after the parents have been advised about the reason.
§2. An infant of Catholic parents or even of non-Catholic parents is baptized licitly in danger of death even against the will of the parents.
Canon 869 §1. If there is a doubt whether a person has been baptized or whether baptism was conferred validly and the doubt remains after a serious investigation, baptism is to be conferred conditionally.
§2. Those baptized in a non-Catholic ecclesial community [that is, generally speaking, Protestants] must not be baptized conditionally unless, after an examination of the matter and the form of the words used in the conferral of baptism and a consideration of the intention of the baptized adult and the minister of the baptism, a serious reason exists to doubt the validity of the baptism.
§3. If in the cases mentioned in §§1 and 2 the conferral or validity of the baptism remains doubtful, baptism is not to be conferred until after the doctrine of the sacrament of baptism is explained to the person to be baptized, if an adult, and the reasons of the doubtful validity of the baptism are explained to the person or, in the case of an infant, to the parents.
Canon 870 An abandoned infant or a foundling is to be baptized unless after diligent investigation the baptism of the infant is established.
Canon 871 If aborted fetuses are alive, they are to be baptized insofar as possible.