The rotor is the "disc" in disc brakes. Disc brake rotors are measured in four primary fashions: thickness, diameter, offset, and pattern. Thickness is the thickness of the portion of the rotor gripped by the brake pads which are loaded into the brake caliper. Diameter is the outer diameter of the rotor. Offset is the distance between the part of the rotor through which the wheel studs typically pass. Pattern is typically the pattern of the wheel studs. Some vehicles have inboard disc brakes, in which case the discs bolt up someplace else, usually to the output yokes of the differential or transaxle.
Rotors are usually made of steel, but carbon fiber rotors have recently gained immense favor in racing due to their light weight and even ceramic rotors are occasionally used in specialty applications.
Ventilated rotors are made of two plates separated by some sort of spacers, usually vanes or studs, to allow air to pass between them and carry away excess heat. Cross-drilled rotors have holes drilled through them in order to increase surface area for the dissipation of heat and in the case of ventilated discs, to allow gases to escape as the brake pad is abraded. Cross-drilled rotors actually have less surface area that the pads can grip, and as such actually have less braking force than solid rotors, but they dissipate more heat and as such disc brake systems which utilize cross-drilled rotors (which can be any of them) are less susceptible to brake fade due to pad heat, and are also less likely to boil the brake fluid which ruins it permanently. Slotted rotors have chamferred slots cut into them, also for the purpose of allowing brake pad exhaust gases to depart without imparting as much of their heat to the rotor as they would otherwise.
Rotors are made in both one and two piece designs. In the case where they are two pieces, the part which the studs pass through is known as the "hat" and the remainder which is gripped by the pads is called the disc.
Rotors which are repeatedly overheated may warp which results in wheel vibration which in turn can prematurely wear tires and wear out components in the steering and front suspension systems. You can identify this condition by pressing the brake pedal successively harder while coming to a stop - if a pulsing is felt (for some reason other than mashing the pedal on a vehicle with ABS to the floorboards) then the rotors are likely warped. In addition, they should be periodically inspected to ensure that they are not too thin. This is carried out with a specialty brake disc micrometer which usually has a flat anvil and a pointed stirrup. The point is placed in the deepest groove one can find on the rotor (usually near the outside) and a measurement taken. It is usually necessary to take multiple measurements around the rotor; significant variation in these measurements usually indicates a warped rotor, which can also generally be detected with a straight edge. Rotors can be machined in a process known as "turning" which can be done on an ordinary lathe or on a brake lathe which may also be capable of "turning" brake drums as well. They typically can be machined only about 1 or 2 millimeters at most before they become too thin and must be discarded and replaced.