What causes a 'stuffy' or 'runny' nose when you have a cold?
It's mainly caused by swelling of large veins and by an increase in vascular permeability that leads to an accumulation of fluids in the nasal mucosa. These effects are mediated, at least in part, by bradykinin and histamine, and can be counteracted by epinephrine. These mediators are part of the immune response to the viral particles.

You can read more about it here:


http://www.cardiff.ac.uk/biosi/subsites/cold/commoncold.html
Chernick-Mellins basic mechanisms of pediatric respiratory disease
By Gabriel G. Haddad, Steven H. Abman, Victor Chernick (page 520)