The term catoptric, is used to describe a telescopic instrument that uses mirrors to collect light; one that uses lenses for that purpose is termed a dioptric, so it should come as no surprise that a telescope using both lenses and mirrors to gather and focus light for magnification by the eyepiece is termed a catadioptric.
The theoretical number of permutations possible when combining variously curved mirrors with variously figured lenses of different densities is (please forgive the pun) astronomical, but there are two types of catadioptric telescopes that have proven to be very popular with serious amateur astronomers world-wide.
Dmitri Maksutov began development of his famous telescope design in October 1941. The Maksutov uses a spherically curved primary mirror that has, in it's Cassegrain format, a central perforation. Light passes through a spherically curved meniscus lens at the top of the telescope which acts as a corrector for the spherical abberation of the mirror. The meniscus lens also has a circular mirrored area in the centre that acts as a secondary mirror and which, due to it's curvature, extends the effective focal length of the primary so that the focal plane of the instrument lies beyond the primary mirror perforation. Maksutov also patented other permutations including a Maksutov-Gregorian and a Maksutov-Newtonian.
The Schmidt-Cassegrain telescope combines the principles of the classic Cassegrain reflector with that of the original Schmidt camera. The Schmidt design used a corrector plate in combination with a spherical primary to produce a curved or spherical focal 'plane' where a curved photographic plate was located. This arrangement gave the camera a very wide field of view. The popular Schmidt-Cassegrain design also uses a corrector plate that is flat on one side and slightly aspheric on the other to remove spherical abberation. The Cassegrain secondary is mounted on the corrector and reflects the light gathered by the primary back through it's central perforation (and extending its focal lenght) for magnification by the eyepiece.