A wet dry filter generally will have bio-balls, filter sponges and / or filter floss type material in it. Or some combination of those. These filters work for removing waste in general, but are maintenence intensive IMO in that you have to clean out the sponges and change out filter materials and so forth. The bio-balls can also cause your nitrates to climb. Some people use them with success, others would never use them. I can't slam them because I've never used one.
A sump is just another container of tank water, unsually under the tank in the cabinet. Water from the main tank is siponed or pumped out of the main tank by use of an over flow box, or a built in overflow. A lot of people drill their tanks so there is no over flow box. Anyway, the water runs out of the main tank into the sump. You can put your skimmer, heater and return pumps under the tank in the sump. Then the water is pumped back into the main tank by a return pump. This way you have those things out of sight, and more room for cool stuff in the tank. How you set up a sump depends on if you want submersible pumps or external pumps.
A refugium is basically another sump with live sand and live rock and macro algae in it. A refugium will help you lower nitrates and keep them lower, as well as give you a safe place for small creatures to grow and reproduce that would ordinarly be eaten in the main tank.
These filter systems can all be used together, any combination or by themselves.
I use a sump / refugium. I don't have any sponges to clean or filter stuff to change out, and it helps to keep my nitrates low. I like to try to keep everything simple. It pretty much comes down to what you want to use. There are a lot of people here that use big rubbermaid containers for a sump / refugium, others use a smaller fish tank or something like that.
NMreef has some good pics of his sump and refugium. Others do to, he's just the only person I can think of right now. He's good for sharing info and ideas and did it all himself if I remember right. Ask around for those that set up their own sumps / refugiums, and how they did it. You'll get a ton of ideas, then just figure out what you like and what fits your budget.
If I messed anything up here, someone will correct me.