The particular product you point us to claims the following:
"Save Up To 16,000 Gallons Of Water Per Year, Per Household (Based on
average modern household usage) And Uses Less Energy Than A 25-Watt
Light Bulb! A Breakthough In Home comfort!"
It's a recirculating pump.
Less energy than a 25W bulb? Under ideal conditions, for the pump
only, maybe. The reason the water at the tap is cold in the first
place is that it can't stay hot in the pipes. Water in the pipes
approaches the temperature of the inside of your walls by losing or
gaining heat through the walls of the pipe. Hot water, obviously,
loses heat into your walls.
The pump puts this now cooler water back into the water heater, where
it can be heated up again. That takes a lot more energy than a 25W
Let's assume that it's always Winter when you're using this
recirculating pump, so you're running your furnace to heat your house.
Let's also assume that it costs no more per BTU for you to heat water
than it does to heat the living space in your house. Finally, let's
assume all the lost heat from your hot water piping escapes into the
living space of your house.
In that case, the energy required to re-heat that cooler water is a
wash--you're just going to run your furnace less and your water heater
Otherwise, just let the hot water run for a minute or two before you
jump in the shower.
The other selling point--16,000 gallons/year? If you have 5 people
showering (2.5gpm) every day, AND it takes 3 minutes for the water to
warm up AND they each start their shower with completely cold water in
the pipes, AND you waste some more water waiting for the sink to get
warm for dishes or shaving...
16,000 gallons/year is a bit of a reach.
If your family is wasting that much water, there's probably a lot of
things you can do to use less of it. Things that don't involve
spending any money at all.