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Refractor, reflector or catadioptric: choosing a telescope design
Every telescope is one of three families: refractors that use lenses, reflectors that use mirrors, and catadioptrics that use both. Each trades cost, size, maintenance and best target differently.
The three families
Refractors use a lens at the front and give high-contrast, low-maintenance views, but cost the most per millimetre of aperture and stay small. Reflectors, chiefly Newtonians and Dobsonians, use mirrors and give the most aperture for the money, at the cost of occasional mirror alignment. Catadioptrics, the Schmidt-Cassegrain and Maksutov-Cassegrain, combine a corrector lens with mirrors to fold a long focal length into a compact, sealed tube.
No single design is best at everything. The right choice depends on what you most want to observe, how much you can carry, and how much you want to spend.
By best target
For the Moon, planets and double stars, a good refractor or a Maksutov gives crisp, high-contrast detail. For faint galaxies, nebulae and clusters, raw aperture wins, so a Newtonian or Dobsonian gives the most for the money. For a compact all-rounder that reaches both planets and small deep-sky objects and travels easily, a Schmidt-Cassegrain is the popular middle ground.
Wide star fields and the Milky Way suit a short refractor or a fast Newtonian, while long focal-ratio designs are made for high magnification on small, bright targets.
By portability and upkeep
A small refractor and a small Maksutov are the most grab-and-go, needing no alignment and cooling quickly enough for casual use. A Dobsonian gives the biggest views but is the bulkiest to move. An SCT is a strong compromise between reach and portability.
Reflectors need occasional collimation and dusting; refractors and sealed catadioptrics need very little upkeep but the catadioptrics take longer to cool to the night air.
Design does not change the eyepiece maths
Whatever the design, magnification is always the telescope's focal length divided by the eyepiece's focal length, the true field is the eyepiece apparent field divided by that magnification, and the exit pupil is the aperture divided by the magnification. Design changes the focal ratio and the field on offer, not the formula, so the fitment tool works the same for any of them.
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Sources: Reflecting telescope (Wikipedia, CC BY-SA) · Refracting telescope (Wikipedia, CC BY-SA) · Catadioptric system (Wikipedia, CC BY-SA) · Telescope Optics, Rutten and van Venrooij; Sky & Telescope optics primers