A sump is a tank underneath the display tank that water drains into and is then pumped back up. In a sump you can have a wet/dry, refugium, skimmer, pumps, heaters, reactors, and many other things. Wet/dry is a type of filtration that is usually in a sump. In what I refer to as a modern reef aquarium a sump is used but not wet/dry, at least not bioballs. People found that if they took out their bioballs the only difference was less nitrate. Some replaced it with live rock, some simply don't use wet/dry at all.
Here is a picture of my sump. On the left you can see the 'bubble section'. This is where all water that goes into the sump is dumped. The overflow box drains into this section, as well as the skimmer, auto top off, and reactors. The middle section is the refugium. There are multiple types of refugia but the usual is used for nutrient export. To achieve this you put lots of 6500K lighting over this section of the sump and grow macroalgae (chaetomorpha is best). The last section is for pumps. The main return pump, skimmer pump, and reactor pumps are housed in this section. To the right of the sump you can see the 2-gallon reservoir for the auto top off.