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Most practical reactors are "thermal" reactors, that is, they utilize the higher thermal cross sections. Possible fuels include some of the various isotopes of uranium (U) and plutonium (Pu).   The only naturally occurring fuel with suitable properties of significant quantities is U-235, hence most reactors use this fuel.  Naturally occurring uranium is composed of 0.7% U-235.  The rest is U-238.  This percentage is too low to sustain a chain reaction when combined with most practical moderators.  Hence either, the probability of fission must be enhanced or the moderator effectiveness must be enhanced.  One group of reactor types (PWR, BWR, HTGR) enrich the fuel (a costly task) and use a cheap moderator (ordinary water or graphite).  In another class, natural uranium (relatively cheap) is used with an excellent but expensive  moderator (heavy water).  This is the CANDU approach.  Which is better?  There is no simple answer.  Both work. In engineered systems, there are always tradeoffs and the final design has to be viewed in the overall context of the end-use environment.
The best moderator to slow down a speedy neutron is something that is the same size as the neutron itself - hydrogen. However, hydrogen does absorb neutrons as well and we want to preserve these precious neutrons so that they can cause fission.  The deuterium isotope of hydrogen, at twice the mass of hydrogen, is almost as good a slowing down agent but, since it already has an extra neutron in the nucleus, it has a very low absorption cross section.  So, overall, deuterium is a far better moderator than hydrogen.  By using deuterium in the form of heavy water, natural uranium can be used as a fuel.  If ordinary water is used, the fuel must be enriched in U-235.  Other possible moderators include graphite and gases such as carbon dioxide and helium.  A good moderator has a high scattering cross section, a low absorption cross section and slows down the neutron in the least number of collisions.
Radiation takes on a number of forms. 
Alpha and beta particles are energetic charged particles that cannot penetrate solids to any significant degree.  So as long as the radioactive fission products are contained by a fuel sheath or some other pipe or wall, there is no concern.
Neutrons are not charged and can penetrate solid walls.  We protect ourselves from them by thick walls that slow down and absorb the neutrons.  Combinations of hydrogenous materials (like water and hydrocarbons) and absorbing materials (like boron and cadmium) make good neutron shields. 
Gamma radiation, essentially very energetic photons (ordinary light is low energy photons), are best stopped by dense material like lead and concrete