Why do antidepressants have a delayed onset of action?
Short answer
The mechanism of action of serotonin-specific reuptake inhibitors, a commonly-prescribed class of antidepressants, is a downregulation of 5HT1A receptors through negative feedback. This downregulation of receptors takes time. 

Background
Serotonin-specific reuptake inhibitors (SSRIs) are a class of antidepressants commonly prescribed to treat depression. Their therapeuric effects may need weeks of treatment (Celada, 2004), which can be explained by their action on 5HT1A receptors in the dendritic region of the serotoninergic neurons in the raphe nucleus. By blocking serotonin (5HT) reuptake, negative feedback mechanisms cause the 5HT1A receptors to downregulate. This downregulation of the presynaptic modulatory 5HT1A receptors changes the firing pattern of the serotonergic neuron into a more fluttering mode, so that it releases more 5HT (Duman, 2007). More 5HT makes you feel better. That is how it fits into the monoaminergic hypothesis. To have the 5HT1A receptors respond this way takes time, as it is an adaptive response that downregulates of 5HT1A receptors, i.e., less receptors are present on the cell's membrane.

This is only an expanation of delayed action of SSRIs, a specific subclass of antidepressants. For example, there are newer classes of antidepressants that seem to work faster (5HT4 action) (Duman, 2007).

References
- Celada et al., J Psychiatry Neurosci (2004); 29(4): 252–65
- Duman, Neuron (2007); 55(5): 712-25