Reef aquariums need ultraviolet light for two purposes. First, corals and other inverts need powerful light, with a UV component, to thrive. They do not need much UV light specifically, but it is a requirement for photosynthesis. Second, UV sterilizers can help slow the spread of dangerous microorganisms, making outbreaks of disease less likely. These two uses for ultraviolet radiation require different equipment.
Ultraviolet light sits on the EM spectrum just beyond blue/violet visible light, hence the name. Too much UV radiation can damage and kill living things, but a small amount can promote photosynthesis. Reef aquariums, by definition, feature corals and other inverts with symbiotic algae that need powerful light with a little UV to produce energy. You can use UV light both to promote healthy reef invertebrates and sterilize the water depending on what kind of UV equipment you use.
Lights designed for a reef aquarium will usually produce the small amount of UV radiation that corals need to thrive. You don't have to go out of your way and buy extra lights to produce UV radiation for your coral. Reef aquariums need powerful lighting anyway, like the light produced by metal halides, advanced fluorescent tubes and LED arrays. For most types of lighting, you want to use at least 5 watts per gallon. However, LED arrays are so energy-efficient, you may need to resort to more complicated measurements. Most store-bought LED arrays come with charts describing their output in PAR, which can be used to figure out how many lights you need. All of these types of lighting produce enough UV light for your corals.
You can also use UV light to sterilize aquarium water. High-end pet shops sell equipment called UV sterilizers that kill microorganisms in the water column. This makes it harder for disease and undesirable algae to get a foothold and spread in a reef aquariums. UV sterilizers cannot totally prevent every case of disease or every patch of algae, but they make it harder for organisms to spread. UV sterilizers are not "required" per se, but can make life easier in the complex world of reef aquariums.
Concerns with UV
Ultraviolet light can hurt you if misused. Always follow the manufacturer's instructions to the letter. If you can see a UV bulb and it is on, something has gone catastrophically wrong. Never power-up a UV sterilizer unless the case is totally closed and you've triple-checked the instructions. Intense UV light can blind you, burn your skin or cause cancer. Another concern, never plumb the return pipe from a refugium through a UV sterilizer, since it can easily kill copepods and amphipods.