There are way too many app icons on my dock. Is there a way to add drawers of apps on the dock so each drawer contains a list of apps?
While the Dock doesn't have the concept of "folders" for the apps that are pinned to or running in it, as of Mac OS X 10.5 Leopard the Stacks feature can provide something similar.
Dragging the Applications folder from the Finder into the "Stacks section" (right of the white-dashed line) of the Dock will add it as a Stack. You can then click the folder to see something like this:
If you want subsets of your Applications as separate Stacks, you can create folders for each somewhere in your filesystem, and add aliases for the desired applications. (Note that dragging an app from the Applications folder to another folder makes an alias by default. Convenient!)
The results look like this:
All this being said, an alternative if you have OS X 10.7 Lion is to use Launchpad to organize and access your apps, keeping the contents of your Dock as minimal as possible.