Catechesis is from Greek katechesis "instruction" (from kata- "thoroughly" + echeein "to sound, resound, talk"). The word as used today means systematic instruction in religious doctrine and practice. There is Jewish, Catholic, Orthodox, Lutheran, and Anglican and Episcopalian catechesis, among others, most often taught in Jewish or Christian parochial schools. Catholic students who cannot receive catechesis in parochial schools enroll in CCD classes (Confraternity of Christian Doctrine). Protestant children are taught in Sunday schools.
Catechesis is given to children and adults. Those families and individual adults who have chosen to convert to Jesus Christ and have accepted Him as Lord and Savior and are receiving instruction in the Christian Faith in preparation for Baptism are anciently and traditionally called catechumens, meaning those who are being catechised (instructed) by catechists (teachers of catechism). The systematic form of catechesis or Christian catechetical instruction is called catechism. An approved study textbook for the course of instruction is also called a Catechism: "Be sure and bring your Catechism". Various editions are prepared for various ages and levels of understanding, from the most elementary all the way to post-graduate levels.
For example, Martin Luther wrote for his disciples the Smaller Catechism and the Larger Catechism. The compendium of Catholic doctrine produced by St. Pius V after the Council of Trent as a standard guide for parish priests is called The Catechism of the Council of Trent and The Roman Catechism. The well-known Baltimore Catechism of 1885 was "prepared and enjoined by order of the Third Plenary Council of Baltimore", and was issued in four editions, #1, #2, #3 and #4, the first one for the elementary grades, the second and third for middle school and high school levels, the fourth one prepared for catechists and priests. The Catechism of the Catholic Church, Second Edition (CCC) was promulgated by Pope John Paul II in accordance with the official Latin text in 1997.
The word catechism has nothing to do with the catacombs of ancient times where Christians secretly met for worship during Roman persecution.