A nuclear fuel rod is a device used to contain radioactive fuel, and its waste products, inside a nuclear reactor. The usage of a rod is a mechanical and engineering decision: it is not absolutely necessary to put the fuel into a rod, but most models of nuclear power plant use some type of rod. The nuclear fuel itself is in the form of a thumb sized pellet. Many of these pellets are put together into a long tube, made of some material that can contain the waste products of the fuel. Currently, the most common material for fuel rods is Zirconium alloy, due to its durability and low neutron absorbance.
The simplest description I can think of is a pringles can: the nuclear pellets being the pringles, and the rod being the pringles tube.
Many groups of rods are combined together into a fuel assembly, which is housed inside a reactor vessel. Although physically separated from each other, the pellets in different rods still effects each other because the containers are transparent to neutrons. (As opposed to the inside of the reactor vessel, which either reflects or absorbs neutrons).
The main job of the fuel rod is to contain the daughter isotopes of the fissioning material. The pure starting fuel, through fission, becomes a host of elements that are some combination of radioactive, poisonous, corrosive or explosive. Some of them are also gases, such as xenon or krypton, that would naturally escape into the atmosphere. The job of the fuel rod is to contain these elements.
After it has used its fuel, the nuclear fuel rod is taken out of the reactor core, and places in a cooling pond, where the radioactivity and heat of the daughter isotopes is allowed to slowly decline.
In the event of an emergency ("meltdown"), the hope is that the nuclear fuel rod will maintain its structural integrity. Often, with the combination of high temperatures, high pressures, and radioactive energy in a nuclear emergency, the thin shielding of the fuel rod ruptures, allowing the stored volatile radioactive gasses into the atmosphere.