Incandescent Bulb Basics
An incandescent light bulb is powered by an extremely thin filament of tungsten wire. As electricity flows through the wire, it heats up. As it heats up, the wire emits most of its heat as infrared radiation. Some of this heat energy, however, is actually emitted as the slightly higher frequency radiation we know as visible light. The connecting wires conduct electricity into the filament. They are deliberately kept relatively cool, since they are not inside the glass globe and unable to shine light into the room.
Resistors and Conductors
The tungsten filament is what is called a resistor--an electric component designed to resist the flow of electricity. That is one of the reasons why it is so thin. The thinner a wire, the harder it is for electricity to pass through and the more resistance the wire poses. The connecting wires, by contrast, are conductors--wires designed to carry electricity with as little resistance as possible. Their only purpose is to transport the electricity into the filament.
Resistance and Heat
Electric current is a form of energy. When that energy meets a resistor, it gets turned into heat. The connecting wires in the light bulb only resist the flow of electricity a little--almost all of the resistance is concentrated in the filament. Therefore, almost all of the heat is generated in the filament. Some heat is generated by resistance in the connecting wires, and some flows from the filament wire into the contacts, but not nearly enough to make the connecting wires glow.