Boron actually mostly acts as a neutron poison, i.e. it absorbs neutrons. That way further (stimulated) decay of other nucleii that were hit by a neutron is eliminated and the chain reaction stops, reducing the production of thermal energy.
A moderator on the other hand inelastically scatters neutrons, converting some of the kinetic energy of the neutrons to heat and leaving a slower neutron. Moderators are used in reactors to bring neutrons down from their initial velocity to velocities where they are more likely to stimulate fission in another nucleus.
When discussing scattering of particles, "elastic" usually refers to electrostatic slingshots where almost the entire kinetic energy of the particle is conserved. Inelastic, however, refers to a direct collision where there is a significant energy loss of the particle. Obviously the former doesn't happen with neutrons, but I believe it's still a typical convention in particle physics.
No, elastic means an encounter where kinetic energy is conserved, rather than being converted to some other form of energy.
THe neutron/nucleus collisions in a moderator are largely elastic. The total kinetic energy of the neutron and the nucleus is conserved, although kinetic energy is transfered from the neutron to the nucleus.
An inelastic collision would involve (for example) nuclear excitation of the target nucleus, or ejection of particles from the target nucleus. The neutrons from fission are usually too low in energy for this to be very likely, though.
A moderator on the other hand inelastically scatters neutrons, converting some of the kinetic energy of the neutrons to heat and leaving a slower neutron. Moderators are used in reactors to bring neutrons down from their initial velocity to velocities where they are more likely to stimulate fission in another nucleus.