Basically it will remove dissolved organics from the water column. That's the quick and dirty......literally. What it pulls out will smell horrible, which would have been left in your water if you weren't running one.
Can a tank be run without a skimmer, sure it can, but it's going to take some other sort of nutrient export to get rid of the bad stuff. A large refugium, bacteria dosing ie: vodka or bio pellets, or more water changes. Using a skimmer is the easiest though.