The moderator of a nuclear reactor is a substance that slows neutrons down. In traditional nuclear reactors, the moderator is the same thing as the coolant: it’s water! When fast neutrons strike the hydrogen atoms in H2O, they slow down a lot (like a billiard ball striking another). There are other good moderators like graphite, beryllium, and more.
When an incoming neutron causes the nucleus of an atom to split, other neutrons are released at very high speed. If at least one (on average) of these neutrons can be made to cause split another fuel atom, a sustained chain reaction is possible. As it happens, fuel atoms (like uranium) are more likely to absorb a neutron whizzing by if it’s going slow (see Figure 1). So, many reactor designs use moderators to make a chain reaction easier to attain.
Some materials are better at slowing down neutrons than others. Conservation of energy and momentum laws explain that a neutron (mass 1) cannot slow down much after a collision with a heavy nucleus, like a fuel atom (mass 235). However, in a collision with Hydrogen (mass 1), a neutron can in fact slow down very far. Imagine a ping-pong ball bouncing off a bowling ball, as opposed to the same ping-pong ball colliding with another ping-pong ball. Clearly, the ping-pong ball will do a better job of slowing things down. So what material has a lot of light elements like hydrogen in it and will make a good moderator? Good old H2O does the trick, giving the name to light-water reactors.
You could mix your moderator and fuel together, but it’s much better to keep them separate. Neutrons can be parasitically absorbed by fuel as they are slowing down in reactions that compete with fission. So if you can have a fast neutron move into the moderator where it will slow down "in peace," i.e. where there are no strong neutron absorbers, and then have it re-enter the fuel as a slow neutron where the probability of fission is extremely high, then it will be easier to make your chain reaction work. This fact was required in the first reactor, CP-1, which used natural uranium because there was no enriched uranium back in the early 1940s.
Thermal neutrons have moderators that allow many neutrons to slow down to thermal energies (i.e. the speed that atoms are vibrating in the surrounding materials due to their temperature) whereas fast reactors don’t have a moderator and their neutrons stay at high energies (i.e. they move fast).