Refractor
The refractor is a long tube type that has a lens in the front. The focuser is on the other end. Refractors generally come in small sizes such as 60mm (2.4") all the way up to 6" or more. The larger the more expensive. There are two types of lens configurations. Achromatic has at least 2 elements.    An achromatic defect in the imaging properties of a lens that results in the formation of colored "fringes" around the image. The problem arises because light of different wavelengths is focused at different distances from the lens. The effect may be countered, but not eliminated completely, by constructing an achromatic lens of two or more components. The other lens for refractors is the color corrected type apochromatic lens. They are superior than achromatic lenses because of the fact they have more than 2 elements and they are desirable for color correction. These type lenses on refractors are more expensive. Refractors of standard quality are very well suited for planetary viewing and astrophotography. Maintenance on refractors are low due to the fact that the lens is fixed to the optical tube. 

The Newtonian reflector (named after Sir Isaac Newton who invented it) uses a parabolic type mirror (bowl shape) to reflect and amplify light sources. In turn the light is directed to a focuser at the front end of the telescope. These reflectors come in a couple of designs. Equartorial and Dobsonian. The equatorial mounted reflectors were the common  type 20 years ago until the introduction of the Dobsonian (Named after John Dobson) which have become more popular due to the simple design and low cost. The Dobs uses a Alt-Azimuth type mount, whereas the equatorial is a tri-axial type mount or "tripod". The equatorial mounts usually have a polar alignment type apparatus.  Also, larger the mirror, the more light gathering power it has. Example: A 8" mirror has about 77% more light gathering power than a 6" and so on. The maintenance on reflectors is more because it may require collimation (mirror alignment) often or when it has been moved roughly. Sometimes it is required to purchase collimation tools if being moved. The mirrors will need periodical cleaning about twice to three times a year. Most reflectors that are larger than 8" tend to be bulky and can be difficult to transport.

The Schmidt-Cassegrain is not a new design. It was designed (but not built) by Jacques Cassegrain in the late 17th Century. It was built after his death. The Cassegrain (SCT) is a catadioptric telescope; which means it employs both a reflective mirror and lens like a refractor. The focusing apparatus either moves the secondary mirror or the objective mirror itself. The main thing the SCT does is that it folds the light to configure a longer focal length in a short distance. Example: A Newtonian 10" at f/9 would have a tube at least 85" to 90" long!  Whereas the SCT at 10" f/9 can be 24" or less. One of the reasons for this is that there is a corrector lens in the front that re-directs light to the objective mirror which in turn directs the light to the secondary which sends the light to the eyepiece. Somewhat complex, but it works. The one thing about SCT's is that they usually are made for a slower focus. They are ideal for deep sky objects, astrophotography and  CCD imaging. Most of these scopes are a lot bigger than they look on magazine ads and catalogs. If you've ever seen an 10" SCT, you'll see what I mean. They can be somewhat massive.

These scopes may require more maintenance  than other type scopes because there are moving parts inside and there are more mirrors to tend to. Most of these scopes come with a fork mount, they are kind of like a alt-azimuth, but with certain accessories (wedges) they can be used as a polar aligned mount. There is a lot involved concerning SCT's, their mounts or tripods are large considering the size of the scopes themselves. They usually require dew sheilds during cooler months because the corrector lens will dew over quite fast. If you decide to do astrophotography or CCD imaging you will need a T-Adapter and focal reducers and such.

Meade & Celestron make some very good quality SCT's that cost between $850-$12,000. Meade has a 16" Cassegrain that is 12 grand and comes with either a bulky tripod or fixed metal stand. The average cost of these scopes is around $2000. My 10" Celestron cost me used $1400. Brand new is about $3100. There are smaller 8" scopes from Meade less than a $1000 that sit on a equatorial mount. I have been to many star parties with people who have SCT's and they love them. Not all SCT's perform the same. The 8" did well, as the 10" did real good. Some folks buy these first, if you do, make sure when you look at the price make sure what comes with it. Many stores only sell the tube assembly only. Or they sell the scope and mount, but nothing else including eyepieces and such.