Listers, there are many kinds of Catholic Churches, and each kind has further subclasses which make for a rather confusing classification system. Here is a quick list comprised of highlights from the Catholic Encyclopedia pages on various kinds of Churches.
1. Cathedral: The chief church of a diocese, in which the bishop has his throne (cathedra) and close to which is his residence; it is, properly speaking, the bishop’s church, wherein he presides, teaches, and conducts worship for the whole Christian community. What properly constitutes a cathedral is its assignment by competent authority as the residence of the bishop in his hierarchical capacity, and the principal church of a diocese is naturally best adapted to this purpose. Such official designation is known as canonical erection and necessarily accompanies the formation of a new diocese. 
2. Basilica: A title assigned by formal concession or immemorial custom to certain more important churches, in virtue of which they enjoy privileges of an honorific character which are not always very clearly defined. Basilicas in this sense are divided into two classes, the greater or patriarchal, and the lesser, basilicas.
- Major: To the former class belong primarily those four great churches of Rome…which among other distinctions have a special “holy door” and to which a visit is always prescribed as one of the conditions for gaining the Roman Jubilee. They are also called patriarchal basilicas, seemingly as representative of the great ecclesiastical provinces of the world thus symbolically united in the heart of Christendom. They possess a papal throne and an altar at which none may say Mass except by the pope’s permission.
- Minor: The lesser basilicas are much more numerous, including nine or ten different churches in Rome, and a number of others, such as the Basilica of the Grotto at Lourdes, the votive Church of the Sacred Heart at Montmartre, the Church of Marienthal in Alsace, &c. There has been a pronounced tendency of late years to add to their number. Besides conferring a certain precedence before other churches (not, however, before the cathedral of any locality), include the right of the conopaeum, the bell, and the cappa magna. The conopaeum is a sort of umbrella (also called papilio, sinicchio, etc.), which together with the bell is carried processionally at the head of the clergy on state occasions. The cappa magna is worn by the canons or members of the collegiate chapter, if seculars, when assisting at Office. The form of the conopaeum, which is of red and yellow silk, is well shown in the arms of the cardinal camerlengo over the cross keys. 
3. Chapel: The basic definition of a Chapel is an informal Church or a room containing a small side altar. There are many different kinds of Chapels. See the NewAdvent page on Chapels for more detail.
4. Oratory: As a general term, Oratory signifies a place of prayer, but technically it means a structure other than a parish church, set aside by ecclesiastical authority for prayer and the celebration of Mass. In the Latin Church oratories are classed as:
- Public: Canonically erected by the bishop and are perpetually dedicated to the Divine service. They must have an entrance and exit from the public road. Priests who celebrate Mass in public oratories must conform to the office proper to those oratories, whether secular or regular. If, however, the calendar of an oratory permits a votive Mass to be said, the visiting priest may celebrate in conformity with his own diocesan or regular calendar.
- Semipublic: Those which, though erected in a private building, are destined for the use of a community. Such are the oratories of seminaries, pious congregations, colleges, hospitals, prisons, and such institutions. If, however, there be several oratories in one house, it is only the one in which the Blessed Sacrament is preserved that has the privileges of a semipublic oratory. All semipublic oratories (which class technically includes the private chapel of a bishop) are on the same footing as public oratories in regard to the celebration of Mass.
- Private: Those erected in private houses for the convenience of some person or family by an indult of the Holy See. They can be erected only by permission of the pope. Oratories in private houses date from Apostolic times when the Sacred Mysteries could not be publicly celebrated owing to the persecutions. Private oratories are conceded by the Holy See only on account of bodily infirmity, or difficulty of access to a public church or as a reward for services done to the Holy See or to the Catholic cause. The grant of a private oratory may be temporary or for the life of the grantee, according to the nature of the cause that is adduced. 
5. Parish Church: A Church under the authority of a priest legitimately appointed to secure in virtue of his office for the faithful dwelling therein, the helps of religion. It must have besides the liturgical equipment necessary for Divine worship, a baptismal font (exception is occasionally made in favour of a cathedral or a mother-church; hence in the Middle Ages parish churches were often called baptismal churches), a confessional, and a cemetery. According to canon law, every church should have a stable income, especially land revenues, sufficient to insure not only the Divine service but also the support of its clergy. 
6. Crypt: An underground church, generally built among the dead. At first, crypts were sometimes as deepsunk as the cubicula of the catacombs themselves. Or they were but partly above ground, and were lighted by small windows windows placed in their side walls. Occasionally their floor was but little below the surface of the ground.