A lightbulb's wire glows hot because, while a large amount of electricity is flowing through, it is narrow, creating resistance (and therefore also heat and light).
Welding also relies on resistance in the circuit, but instead of resistance in the circuit being caused by a sudden narrowing of the path, it is created by the crossing of a gap. The gap is kept small enough so that electricity is able to jump across it, completing the circuit and creating the arc. The temperature of the arc has been measured at 10,000 degrees F - almost as hot as the surface of the sun.
