Like making ICs. You start by putting on a photoresist, typically by spin coating. Next, a concentrated light (eg: laser or mask) exposes (or doesn't expose depending on the type of photoresist) the area that you want to etch. Finally, the area is chemically etched away.

Photoresists http://cleanroom.byu.edu/photoresists.phtml

A high power pulsed laser is fired at the area to be removed. Professionals seem to lean towards this for its ease of use and repeatability.

A microprobe scrapes the traces away. Professional units use ultrasound to make the cutting more reliable.

Tarnovsky seems to favor using nail polish to form a mask. He then scrapes the to be removed area's mask away with a probe and chemically etches. His results indicate this works surprisingly well despite how crude it sounds vs other techniques.

Some fuses are blown by sending pulses. Not practical in most cases since the area to cut could be very obscure.