Almost nothing, under every day circumstances
Our atmosphere blocks out almost all UV light - this is why the other answers are talking about shining UV light on things.
Q: If you get a UV camera and just point it out your window, what does it pick up?
A: Practically nothing - there will be more input from electrical noise in the camera affecting the sensor than there will be UV input, unless you point it directly at the sun.
Your character will be able to see UV sources directly, without needing things to flouresce to detect its presence.
electrical arcs, blacklights, rocket motors etc would be more visible to them.
Wikipedia notes that some people can see UV light
Ultraviolet rays are invisible to most humans. The lens of the human
eye blocks most radiation in the wavelength range of 300–400 nm;
shorter wavelengths are blocked by the cornea. Humans lack color
receptor adaptations for ultraviolet rays. Nevertheless, the
photoreceptors of the retina are sensitive to near-UV, and people
lacking a lens (a condition known as aphakia) perceive near-UV as
whitish-blue or whitish-violet. Under some conditions, children and
young adults can see ultraviolet down to wavelengths of about 310
For Near UV light vison, this will be like having near IR vision - like a cheap camera such as you will find in your phone or as a webcam has - it'll pick up on e.g. a remote control for a TV (which uses near IR) as a glow, but it can't see beyond that - in the IR space that would be thermal vision.
There still isn't much UV light that makes it to the surface of the earth. These people are able to see the light from UV sources primarilly.
For UV, being able to see beyond near UV light, you're passing out of the range covered by blacklights to the shorter wavelengths, that are absorbed by the Ozone layer.
Ozone absorbs more than 99 percent of UV-C rays -- the most dangerous portion of the spectrum. Ozone absorbs about 90 percent of the UV-B rays
you're not going to see much at all in this region, even if you block out all of the visible light.
The main reason we have evolved to see the visible spectrum is because that range is mostly transparent to our atmosphere - the light in that range reaches us. Having a larger visual range is not beneficial, as the intensity drops off rapidly: