Lenses (refractor) or mirrors (reflector) telescope? What differentiates, in terms of practical quality, not technical implementation, a refractor from a reflector telescope? 
Why would one prefer a refractor over a reflector, when reflectors come with such large diameters at a smaller price?
 A: Lenses are expensive, and have an upper limit on their size (when they get too big they get really expensive, and you start seeing a lot more aberration effects). Really cheap telescopes are usually refractors (at least from what I've seen), but if you want a respectable telescope, refractors climb in price rather quickly, in my experience.
Reflector telescopes are more common, smaller in size, generally less expensive and offer higher quality pictures without some of the aberration that comes with a refractor.
A: First of all, Carson is correct in that refractor lenses climb exponentially in price as the size of the lens grows.
Reflecting telescopes are very useful for astronomy, and it's easy to spend a small amount of money and end up with a hefty tube that can gather a ton of light with very few drawbacks.
On the other hand, a high-quality small to midsize (and thus more transportable) refractor can more-easily perform many other useful duties such as daytime photography, easier viewing of land-based targets, and serving as an all-around travel (or spontaneous use) scope. 
Apochromats, which feature some of the best lenses among refractors, have fallen in price considerably in recent years, so it's now possible to obtain a very versatile, top-quality refractor in the 60 to 110mm range. Beyond that size, many of these tasks become much more difficult and require increasingly sophisticated mounting setups -- this is especially true of photography.
Most amateur astronomers I've talked to recommend switching to a reflector at that point, because the primary use of larger scopes at that point becomes astronomy, and it is here that reflectors have a great bang-for-buck.
A: Refractors suffer from fewer optical aberrations than reflectors because they have only two elements in their optical assembly, making it easier to align and maintain collimation.
The biggest problem with refractors is chromatic aberration, which can be corrected to a certain degree by Apochromatic lenses, but isn't completely eliminated.
For deep sky objects which are visually faint, chromatic aberration is not a significant problem, and hence refractors may be preferred.
For solar system objects and brighter DSOs, the chromatic aberration might become significant.
A: Reflector versus Refractor has been tossed back and forth for a very long time.  The refractor is better than a reflector beginning with the issue of collimation.  Refractors are constructed in such a manner that the doublet on the front of the scope has "set" spacing. There is either an air gap or an oil gap.  The size and quality of the objective lens determines the cost.  AstroPhysics makes a six inch VERY high quality refractor.  Other manufacturers to consider are William Optics and Televue for smaller size ones. Chromatic abberation is NOT common complaint amongst owners of these products made by the formerly mentioned makers.
Reflectors, usually referred to as Newtonians, have a constant collimation issue every time you move them any real distance.  Astrophotography is just as possible optically with either refractors or reflectors.  The second cost factor is the telescope mount.  A German Equatorial Mount, (GEM), is the only GOOD way to track any celestial object with accuracy and elimination of field rotation.
There are only one or two other types of optical systems beside these commonly available. One of the most common is the Schmidt Cassegrain telescope or SCT.  It is a specialized type of reflector that is focused by moving the mirror forward and back within the optical tube assembly or OTA.  The eyepiece views through a hole in the center of the spherical mirror.
A: There are actually three main telescope types, refractor, reflector, and catadioptric (which uses both lenses and mirrors to form its image). Each has its advantages and disadvantages, which is why all three are manufactured currently.
From a practical standpoint, refractors offer the sharpest and most contrasty images in a given size, because of their unobstructed apertures. Reflectors offer less sharpness and contrast because of the presence of a diagonal mirror and its supports in the middle of the main optical path. This can be minimized by reducing the size of the obstruction to under 20% of the aperture (by diameter). Catadioptric telescopes offer the poorest sharpness and contrast because of their more complex optical systems, which also almost always include a large central obstruction. In return, they provide extremely compact designs which are easier to transport and mount. Catadioptrics using a Maksutov corrector tend to perform better than those using a Schmidt corrector because their central obstructions are smaller and their correctors use simple spherical curves.
Part of the advantage of refractors is countered by their chromatic aberration. All refractors suffer from this to some extent, leading to colour fringing around bright objects. It can be reduced in two ways: going to a long focal ratio (f/15 or longer) or using special low dispersion glass (ED or Extra-low Dispersion or fluorite). These solutions each have their own disadvantage: telescopes with long focal ratios are difficult to mount solidly, so require large expensive mounts; telescopes with ED or fluorite lenses are often very expensive.
For most amateur astronomers, a high quality reflector offers the best compromise in terms of cost and aperture.
A: Lots of good advise from others. From a practical perspective (I own both a reflector as well as an achromatic and apochromatic refractor) there are a few more thing worth stressing:
1) for a given aperture a small quality refractor offers the best image with the highest contrast and does so without any fiddling: no collimation required, and no waiting for the scope to cool down and the image to stabilize. 
2) Small refractors can be amazingly portable. What is your main observation site (car trip required?) will often determine what is your best scope.
3) Another important aspect is the answer to the question: what objects do you want to observe? People who are happy to view brighter objects are often best off with a small, portable refractor. If you are intested in 'faint fuzzies', you need large aperture (and an observation site low in light pollution). 
Many amateur astronomers start with a Dobsonian reflector as these provide by far the most affordable aperture. Many later buy a small refractor as a second "grab-n-go" scope. In the end, the scope you use most often is your best scope. 
