Catadioptric telescopes
In Maksutov (pictured above) designs the light enters a thick meniscus correcting lens with a strong curvature. The light then strikes the primary mirror and is reflected back up to the secondary mirror that reflects the light out an opening in the rear of the instrument. The secondary mirror is usually an aluminised spot on the back of the meniscus corrector. The Maksutov secondary mirror is usually smaller than the Schmidt's thus giving the Maksutov better resolution for planetary observing. The Maksutov is usually heavier than the Schmidt and the thicker correcting lens takes longer to reach thermal stability.
In Schmidt (pictured above) designs the light enters a thin aspheric Schmidt correcting lens. The light then strikes the primary mirror and is reflected back up to the secondary mirror that reflects the light out an opening in the rear of the instrument. Schmidts usually have shorter focal lengths thus making them more suitable for fainter deep sky objects. The thinner corrector plate means the Schmidt is faster to reach thermal stability.
Advantages
Best all-around, all-purpose telescope design. Combines the optical advantages of both lenses and mirrors while cancelling their disadvantages.
Excellent optics with razor sharp images over a wide field.
Excellent for deep sky observing or astrophotography with fast films or CCDs.
Very good for lunar, planetary and binary star observing or photography.
Excellent for terrestrial viewing or photography.
Focal ratio generally around f/10. Useful for all types of photography. Avoid faster f/ratio telescopes (they yield lower contrast and increase aberrations). For faster astrophotography, use a Reducer/Corrector lens.
Closed tube design reduces image degrading air currents.
Most are extremely compact and portable.
Easy to use.
Durable and virtually maintenance free.
Large apertures at reasonable prices and usually less expensive than equivalent aperture refractors.
Most versatile type of telescope.
More accessories available than with other types of telescopes.
Best near focus capability of any type telescope
Disadvantages
More expensive than Newtonians of equal aperture.
It is not what people expect a telescope to look like.
Slight light loss due to secondary mirror obstruction compared to refractors.
May suffer from image shift when focusing due to focus design moving the entire primary mirror.
May have a narrower field of view (Maksutovs) when compared to similar aperture Newtonians or Refractors due to longer focal lengths.