G'Day Darren,
I'll jump in and I'm sure others will correct me when I'm not so correct!
Most refractors are a "straight through" type of arrangement - light goes in the front and comes out the back, basically. As mentioned before there are achromats, semi-apochromats and apochromats, which are successively more complex (and usually better quality) lenses to reduce chromatic aberration.
There are heaps of different reflector designs, but the main ones are:
Cassegrain - light comes in the front, reflects off a parabolic mirror at the back (the primary) then of a much smaller secondary mirror at the front and finally comes out through a hole in the centre of the primary mirror.
The Scmidt-Cassegrain (SCT) is a variation on this which includes a corrector plate (a lens) at the front to correct some optical defects in the mirror shapes. SCTs are usually either fork mounted or equatorially mounted.
Newtonian - light comes in the front, reflects off the primary at the back, then off the much smaler secondary mirror at the front which is at 45 degrees to the axis if the tube, so the light exits out the side near the top. Newts are most often mounted equatorially or in dobsonian bases.
A Schmidt-Newt once again has a corrector plate at the front.
Hope this helps,
Al.