In One Sentence:
A control rod is a cylinder of material with a large neutron absorption cross section, used to control fission reactors by limiting the number of free neutrons feeding the chain reaction.
Background
A nuclear fission reactor produces electricity with a controlled nuclear chain reaction — uranium 235 or plutonium 239 nuclei split when bombarded with neutrons, and the split nucleus then emits more neutrons which go on to split other nuclei. Because the split uranium nucleus emits more neutrons than it takes to split it, the chain reaction increases in intensity with each iteration of the process. Left uncontrolled, the reaction would continue to grow, getting hotter and emitting more radiation, until something finally breaks. This is what happened at Chernobyl.
Control
To control a fission chain reaction, it is necessary to limit the amount of free neutrons flying around. This is accomplished with a control rod made of a substance which is good at absorbing neutrons into its nucleus, capturing them so they cannot feed the chain reaction. In a process called shimming, the reaction is weakened by pushing the control rods deeper into the reaction chamber, and strengthened by pulling them farther out. For safety purposes, control rods are lowered from above so that in the event of an accident, they naturally fall into the reactor and stop the reaction. Old designs were lowered in by chains or cables, and modern designs are suspended by electromagnets.
Nuclear Cross Section
The more protons and neutrons an element's nucleus has, the larger this nucleus is. Logically, the larger the nucleus is (that is, the larger its cross-sectional area is), the larger a target it is and the more likely it would be for a randomly passing particle to hit it.
It isn't quite that simple though, and as it turns out different elements are better or worse than others at absorbing passing neutrons, with little to do with their actual, physical cross-sectional area. The actual effective target area a nucleus presents is called the neutron absorption cross section, and it varies enormously from element to element and even isotope to isotope of the same element.
The analogy would be how large the nucleus would have to be to account for the observed percentage of particles of negligible diameter hitting the nuclei of the atoms a target is made of. Actual nuclear cross sections are on the order of 10E-24 cm2 (a unit called a barn), and there is a lot of empty space between nuclei. Combined with the fact that a neutron, which has no charge, is neither attracted nor repelled by protons or electrons, it would seem that the likelihood of a collision is very small. In fact, neutrons tend to pass through most materials easily, giving the vast majority a nuclear cross section of less than 100 barns. Most nuclei have a nuclear cross section of less than 1 barn. Gold, for example, has a cross section of 98.65 barns, and lead 207 a mere 0.699 barns.
Two excellent materials for making control rods are cadmium 113 and boron 10 because these particular isotopes have a large neutron absorption cross section. Cadmium 113 has a whopping nuclear cross section of 20,600 barns! Boron 10 is no slouch either with a cross section of 3,835 barns. When a neutron is captured by one of these materials, it undergoes a nuclear reaction of its own, but not a chain reaction since it does not emit any neutrons (which would defeat the purpose). Boron 10 is split into lithium and helium when it absorbs a neutron, and cadmium 113 simply absorbs the neutron to become cadmium 114, emitting only a gamma ray. These materials created by the nuclear reaction are not good neutron absorbers (cadmium 114 has a cross section of only 0.34 barns), and further neutrons simply bypass them to be absorbed by the as yet unreacted material behind it. Therefore, a control rod can only absorb as many neutrons as it has atoms of absorbing material before it is used up.
The Reaction:
113Cd + n = 114Cd + γ
Cadmium 113 has an atomic weight of 112.904400 amu, and is made up of protons, neutrons, and electrons massing a total of 113.9388537 amu. This gives it a mass defect of 1.034453656 amu and binding energy of 963.5841586 MeV (8.527293439 MeV per nucleon).
When a cadmium 113 nucleus absorbs a neutron, it becomes cadmium 114 with an atomic weight of 113.903357 amu, made up of protons, neutrons, and electrons massing a total of 114.9475187 amu. This gives it a mass defect of 1.044161656 amu and a binding energy of 972.6270721 MeV (8.531816422 MeV per nucleon).
The difference in binding energy is 9.0429135 MeV, which is emitted in the form of a gamma ray.
Sources:
http://www.ncnr.nist.gov/resources/n-lengths/list.html
http://www.tpub.com/content/doe/h1019v1/css/h1019v1_113.htm
http://www.wordiq.com