The slip-joint style connection is most common for kitchen sinks, bathroom sinks, and tubs(bath/shower combinations).
The first forms of this style connection were chrome plated brass. They can still be bought in chrome for
decorative purposes, but now the cheap and reliable pvc versions are most common.
A slip joint fitting gets assembled in this manner(picture to the left), putting the nut, then washer on the pipe first. The taper of
the washer should face the fitting(skinny side put into the fitting). Then, when the nut gets screwed onto the fitting,
the washer gets compressed to form the seal.
Most of the time, just tightening the nut hand-tight will suffice, but just a bit more from a set of pliers, about
1/8-1/4 turn, is generally a good idea.
There are a variety of fittings and pieces available, but only a few standard kits. There are kits for double bowl
kitchen sinks, center outlet(the top picture, the bowls meet in the center) and also an end outlet(both bowls connect below one bowl).
All fixtures with slip joints should also be using what is called a trap.
The picture below on the left is a p-trap. The one on the right is an s-trap.
The kits above are standard complete professional kits. Your local hardware store kits may contain more or less pieces.
You'll notice in the pictures above, one washer that is different than the rest. It appears thicker. This is because
there are two different sizes of slip joint traps. 1 1/2" traps are standard these days, but 1 1/4" traps were common years
However, today's bathroom sinks are almost always 1 1/4" coming off of the sink(pop-up assembly). This is the reason for
the extra washer.
This washer is the same outside diameter, but smaller on the inside in order to fit an 1 1/4" pipe.
Slip joint fittings are normally very reliable. They can, however, be subject to coming apart if hit with objects
normally kept under a sink. Depending on the piping, some pieces of pipe could be short or being held together with the
slip joint. In circumstances like this, some joints can tend to come apart. In these cases, the piping needs to be
altered to prevent this. Slip joints are only designed to seal against water, not support weight.