What is a wedged through tenon and how is it constructed? How is it different than a "standard" mortise and tenon? When should it be used?
This is where you mortise right through the timber, and hammer wedges into the joint from the other side, in order to mechanically wedge the tenon into the joint. You still use glue as you normally would.
See here for examples.
I'll include the images here for future reference:
They're typically used when making doors, and are good because they form a good strong joint, and they look nice from the front (better than a mitre joint), however they're not all that nice to look at from the edges, and the timber of the mortised piece can shrink or swell around the tenoned end or the wedges and leave them sunken in or sticking proud after some time. You'll also need to play around a bit with the width of the tenon vs the mortise and the taper angle of the wedges etc. the first time you make one, to get it just right.
Note that (as with just about any type of joint) there are many variations for different situations. See here for many more examples. Probably the most important and commonly used variation is the "haunched" joint which is where you cut part of the tenon away, to space it away from a corner (e.g. the top or bottom of a door leaf):
Adding to WhatEvil posted, there is another type of wedged through tenon, more precisely referred to as a wedged tusk tenon or a tusk tenon. This one is primarily used for knockdown construction. It has many of the same strengths as other mortise and tenon joinery but doesn't use glue. The "pin" (tusk) length also helps a little to prevent racking.
Sometimes there is a double pin, where the pins are horizontal and you use one from each side to hold the joint in place.