An ultraviolet-visible spectrophotometer is a device used to analyze the degree to which a given sample absorbs different spectra of light - as the name suggests, the machine functions in the visible and ultraviolet wavelengths.
In other words, they tell you what color a sample is, and to what degree, out to a large number of decimal points.
The only type of UV-VIS machines I have experience with are diode arrays. These work by shining light through the sample - which is in solution - at different wavelengths, and measuring the amount picked up by a collector on the other side of the sample.
It generates a series of data points relating wavelength to absorption number; the absorption number is a relative unit; all the units put into the calculation cancel.
UV-VIS spectrophotometry is useful for things such as titrations, because the machine can identify a much smaller change in the color of the pH indicator than is visible to the human eye. As an example, consider a lab from my biochemistry class: we were studying enzyme kinetics, and we were using a yeast hexokinase enzyme that catalyzed the phosphorylation of six-membered sugars. The byproduct of this reaction is the release of a proton, one per sugar molecule phosphorylated. The change in the proton (hydronium ion, actually, to be pedantic) concentration can be observed using a pH indicator such as cresol red, whose color change is measured with the UV-VIS by tracking the change in absorption at a constant wavelength (at physiological pH ranges, 560 nm seemed to work well for cresol red). The results are calibrated to the color change produced by adding a known amount of HCl or another strong acid to a solution of cresol red in the same concentration.
They are also useful for measuring the UV absorbance of a sample, which we would be unable to see with our naked eyes anyway; this can be useful for determining the concentrations of protein vs. DNA in solution. | 
