Why does trade depress relative wages? I've read that increased trade tends to decrease relative wages of labor. What is the intuition behind this? Lower wages in labor imply lower demand for workers, but doesn't trade just encourage specialization, meaning that workers can just be hired by whatever industry they decide to specialize in?
The theory is that if you have three types of production sectors in the economy:


Sectors which could export more with liberalised trade
Sectors which could import more with liberalised trade
Sectors which could see little change with liberalised trade


then increased trade could tend to increase wages in the first type of sector, reduce wages in the second type, while leaving the third type less affected.  In addition, greater efficiency associated with liberalised trade could tend to reduce overall consumer prices, boosting the real value of all wages.

So for an economy which has a comparative advantage in low-wage sectors, this theory might suggest that the relative wage gap could narrow with liberalised trade.  By contrast, the same theory might suggest that the relative wage gap could widen with liberalised trade for an economy which has a comparative advantage in high-wage sectors.    

This is only a theory and would require empirical confirmation.  Other things may be happening at the same time, such as changes in the local skill-sets or liberalised labour and consumer markets, which mean that the theoretical effects of liberalised trade are overwhelmed by other effects and the predicted relationship may not be seen.

An example of this sort of discussion can be read at https://wol.iza.org/uploads/articles/378/pdfs/how-does-international-trade-affect-household-welfare.pdf