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There are basically 3 types of astronomical telescopes:
(also known as dioptrics) are what the average person identifies with the word "telescope", a long, thin tube where light passes in a straight line from the front objective lens directly to the eyepiece at the opposite end of the tube.
Advantages
- Easy to use and reliable due to the simplicity of design.
- Little or no maintenance.
- Excellent for lunar, planetary and binary star observing especially in larger apertures.
- Good for distant terrestrial viewing.
- High contrast images with no secondary mirror or diagonal obstruction.
- Color correction is good in achromatic designs and excellent in apochromatic, fluorite, and ED designs.
- Sealed optical tube reduces image degrading air currents and protects optics.
- Objective lens is permanently mounted and aligned.
Disadvantages
- More expensive per inch of aperture than Newtonians or Catadioptrics.
- Heavier, longer and bulkier than equivalent aperture Newtonians and catadioptrics.
- The cost and bulk factors limit the practical useful maximum size objective to small apertures
- Less suited for viewing small and faint deep sky objects such as distant galaxies and nebulae because of practical aperture limitations.
- Focal ratios are usually long (f/11 or slower) making photography of deep sky objects more difficult.
- Some color aberration in achromatic designs (doublet).
- Poor reputation due to low quality imported toy telescopes; a reputation unjustified when dealing with a quality refractor from a reputable manufacturer.
(also known as catoptrics) usually use a concave parabolic primary mirror to collect and focus incoming light onto a flat secondary (diagonal) mirror that in turn reflects the image out of an opening at the side of the main tube and into the eyepiece.
Advantages 
- Lowest cost per inch of aperture compared to refractors and Catadioptrics since mirrors can be produced at less cost than lenses in medium to large apertures.
- Reasonably compact and portable up to focal lengths of 1000mm.
- Excellent for faint deep sky objects such as remote galaxies, nebulae and star clusters due to the generally fast focal ratios (f/4 to f/8).
- Reasonably good for lunar and planetary work.
- Good for deep sky astrophotography (but not as convenient and more difficult to use than Catadioptrics).
- Low in optical aberrations and deliver very bright images.
Disadvantages 
- Generally not suited for terrestrial applications.
- Slight light loss due to secondary (diagonal) obstruction when compared with refractors
use a combination of mirrors and lenses to fold the optics and form an image. There are two popular designs: the Schmidt-Cassegrain and the Maksutov-Cassegrain. In the Schmidt-Cassegrain the light enters through a thin aspheric Schmidt correcting lens, then strikes the spherical primary mirror and is reflected back up the tube and intercepted by a small secondary mirror which reflects the light out an opening in the rear of the instrument where the image is formed at the eyepiece. Catadioptrics are the most popular type of instrument, with the most modern design, marketed throughout the world in 3 1/2" and larger apertures.
Schmidt-Cassegrain Advantages
- Best all-around, all-purpose telescope design. Combines the optical advantages of both lenses and mirrors while canceling their disadvantages.
- Excellent optics with razor sharp images over a wide field.
- Excellent for deep sky observing or astrophotography with fast films or CCD's.
- 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 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.
Schmidt-Cassegrain 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.
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