Why does soap not lather in hard water.

Soap is usually made from one or more fatty acids. Fatty
acids are relatively long chain organic molecules that have both a hydrophobic (water
repelling/fat loving) part and a hydrophillic (water loving/fat repelling) part which is
the carboxylic acid end group. When the acid group is neutralized by combination with a
group one metal ion, the resulting  ion will dissolve in
water.

The way that the soap works is that the hydrophilic
end will interact with the oils and greases that you are trying to remove from your
hands/clothes/etc. and dissolve them.  At the same time the hydrophilic end makes the
molecule water soluble so when you rinse your hands the soap molecule and the dirt are
washed down the drain.

Soaps when added to water also
reduce the surface tension of water, allowing the formation of the lather that you
referred to.  There is a certain minimum amount of soap - the critical micelle
concentration - that is necessary for the lather to form.  That is why if you use soap
with really dirty dishes the initial lather disappears, even in soft
water.

Now, in hard water there are relatively large
amounts of calcium and magnesium ions. Each of these ions can react with two soap
molecules, forming an insoluble precipitate.  Since the soap has been effectively
removed from solution there is not enough molecules left to form the soap lather. The
common  way hard water is noticed is the ring that is left around the bathtub  when
bathing in water that is hard.