Does water absorb neutrons? Water is used as a shielding material in nuclear reactors. What's its function as a shield? Does it absorb neutrons and is there a reaction between water and neutrons?
 A: Neutrons don't see water, but instead see hydrogen and oxygen.  The relevant cross-sections for "thermal" neutrons are
$$
\begin{array}{rccl}
& \sigma_\text{scatter} & \sigma_\text{capture} & \text{(in barns)}
\\
\text{hydrogen} & 82 & 0.33
\\
\text{oxygen} & 4.2 & 0.000\,19
\end{array}
$$
So you can see that the primary effect of the water is for the neutrons to scatter off of the hydrogen.  This exchange will tend to bring the neutrons into thermal equilibrium with the water (at which point we can actually refer to them as "thermal" neutrons, with typical kinetic energy $kT \approx 25\,\rm meV$).
From the neutron's perspective water is a gas of hydrogen atoms with number density 
$$
n = \rm\frac{1\,gram}{cm^3} \cdot \frac{1\,mole\,H_2O}{18\,gram} \cdot \frac{2\,mole\,H}{1\,mole\,H_2O}
=0.11\rm\,\frac{mole\,H}{cm^3}.
$$ The oxygen is a 2% correction the scattering and completely irrelevant for capture.  The probability of traversing a path length $\ell$ without an interaction is $\exp -n\sigma\ell$, so the mean free path between scatters for neutrons in water is roughly
$$
\ell_\text{scatter} = \frac{1}{n\sigma_\text{scatter}} \approx \rm2\,mm
$$
and between captures is roughly
$$
\ell_\text{capture} = \frac{1}{n\sigma_\text{capture}} \approx 450\rm\,mm.
$$
This suggests that a few decimeters of water is pretty efficient at moderating fast neutrons down to room temperature, but a few meters are required to completely convert thermal neutrons into gamma rays due to capture on hydrogen (energy 2 MeV).  A few meters of water is pretty efficient at turning gamma rays into heat.
A: Water serves for water reactors in up to 3 roles: as a moderator, as a coolant and as shielding.
Water is excellent shield for alfa radiation of transuranic elements, -like any condensed matter, decreasing radiation to half in probably less then 1 mm thickness.
It is very good shield for beta electron based radiation, mainly from fission products and neutron decays, with half-thickness typically about 1-2 cm.
It is not good shield for gamma radiation, half-thickness more then 20 cm.
It is bad shield for neutrons in sense of low absorption cross-section,
but it is very good shield - as all hydrogen rich matter - in sense of slowing neutrons down to thermal speed random movement. Those not finding their absorption fate in fission or other material decay to proton, electron and e-antineutrino with a halftime about 14 min.
Products of water neutron absorption is deuterium for light hydrogen, resp. tritium for deuterium ( heavy water or minor D content in natural water ).
1H has better moderating slowing down effect than deuterium, due better matching proton - neutron masses. But unfortunately, 1H has much larger absorption cross section for neutrons then deuterium. Therefore heavy water can be used for reactors using unriched natural uranium - see PHWR and CANDU rector, but light water reactors need to use enriched uranium.
A: Water isn't a particularly effective radiation shield, and as far as I know it is not used as a shield in the reactors themselves.
However it is used as a shield in the pools where fresh nuclear waste is stored. These are known as spent fuel pools, and they use large thicknesses of water (around 10 metres) to absorb the radiation emitted by the fuel.
A: Water acts as a coolant in nuclear reactors. It is run through pipes in and around the reactor to absorb the immense heat generated. 
