How is a trace formed?
Let's examine the manufacturing steps for a very simple single-sided PCB. 1

PCB fabrication is a subtractive process.  The PCB starts life as a solid sheet of copper laminated on a sheet of insulator 2.  Most of the time, traces are formed by etching.   But how to etch away the excess copper between the traces, while keeping the copper that forms the traces?


Copper is covered with photoresist.
Parts of the board that will remained covered by copper (e.g. traces) are exposed to light.
The board is washed with a solvent.  Unexposed photoresist is washed away.  Exposed photoresist remains.  So we've got a board [still] completely covered with copper.  Traces are drawn on top of the copper with photoresist.  
The actual etching takes place.  


The etching solution attacks copper.  
At the same time, the chemistry of the photoresist is such that the etching solution doesn't attack it.  As a result the traces that are made of copper aren't etched away.

Exposed photoresist is washed away with another solvent (different than the one used for unexposes photoresist), which doesn't attack copper.


Thus, we get a board with copper traces.

1 This question is about traces, and I'd like to focus on the traces.  The sequence of steps would be somewhat different when there are have multiple copper layers and vias have to be plated.

2 Typically fiberglass.  If the PCB has to be extra cheap, phenolic paper can be used .

3 Sometimes during prototyping traces can be made by milling.