A wet/dry system has parts that are moist and can be colonized by nitrifying bacteria, but also come in contact with oxygen in the air. A wet/dry system can be set up in a sump, but can also be elsewhere. For example there are canister filters that incorporate a wet/dry system. The bio-wheel on some power filters is a kind of wet/dry system that is not in a sump, and has no refugium associated with it.
A sump is a water reservoir below the main tank that is fed through overflows. A sump can serve many purposes, some of which are filtration, housing of heaters and chillers, use as refugium, set up of wet/dry systems, housing of buffers like crushed coral and peat, and so on. A great resource about sumps is here. The sump of my 240G contains neither a wet dry system, nor is it used as refugium.
A refugium is an area separated from the main tank to contain the same water but prevent predation by the inhabitants of the main tank on the inhabitants of the refugium. A sump can be used as refugium, but a refugium can also be created by using a tank divider. Refugiums can house many things, some of which are plants, fry, invertebrates, and so on. Refugiums are particularly popular with salt water tanks. A breeding box floating in a tank would be a refugium that is not in a sump, and has no wet/dry system associated with it.
Breeding box with little fish (source).
A sump is a water reservoir below the main tank that is fed through overflows. A sump can serve many purposes, some of which are filtration, housing of heaters and chillers, use as refugium, set up of wet/dry systems, housing of buffers like crushed coral and peat, and so on. A great resource about sumps is here. The sump of my 240G contains neither a wet dry system, nor is it used as refugium.
A refugium is an area separated from the main tank to contain the same water but prevent predation by the inhabitants of the main tank on the inhabitants of the refugium. A sump can be used as refugium, but a refugium can also be created by using a tank divider. Refugiums can house many things, some of which are plants, fry, invertebrates, and so on. Refugiums are particularly popular with salt water tanks. A breeding box floating in a tank would be a refugium that is not in a sump, and has no wet/dry system associated with it.

Breeding box with little fish (source).