It varies all over for specific duties, but the basics are they drive the engine to the scene, place the rig and pump water.
The ladder or "truck" driver is similar, except operate the aerial ladder.
For our dept the driver (engineer) is responsible for ensuring the rig is operational, lights work, valves open and close easily, engine fluids are OK, and just to make sure everything works.
On a fire, the first engine on scene will most likely be pumping so the engineer puts the pump in gear, get water going to the fire crew, hook up the large diameter hose from a hydrant, or have a water supply if drafting, and then just basically make sure there is enough water for the lines being pumped.
If the engine is second or third in, then the engineer becomes part of a firefighting crew and could be doing any number of fireground duties.
Like I said this is the very basics and operations vary depending upon department.
Ours is pretty much like what has already been said. The driver drives and operates the engine and pump. In our dept. the driver isn't supposed to even leave the truck unless called for by the IC.