Pumps have an air chamber (normally cylindrical) that is compressed in order to force the air into the tyre. When the handle is pulled out, air behind the piston is forced out of the cylinder. In single action pumps, this air is vented into the atmosphere and lost (no big deal; there's plenty of it around!). On a double (or dual) action pump, this air is collected and forced through the valve into the tyre.
Single action pumps are mechanically simpler, and so smaller and lighter than equivalent double action pumps, but take roughly twice as many strokes to get to the same pressure. With a double action pump, you have to do work when pulling the pump handle, as well as when pushing, which is more effort per stroke (due to the pressure of the air already in the tyre, unlike in the single action pump, where you're working against atmospheric pressure which is much lower).