Yes, U-V vs. V-J just means to compare the difference in magnitudes between U and V bands with the difference between V and J bands. And it's one good way of classifying galaxies.
To understand why, let's start by thinking about stars. As main sequence stars become more massive, they become hotter. You can reasonably approximate the spectrum of a star as a blackbody, which means that as it becomes hotter, it also becomes bluer.
http://zebu.uoregon.edu/~imamura/122/images/blackbody_curves.jpg
Magnitudes are logarithmic, so when we take U-V, we're looking at the flux ratio, or basically the slope between U (ultraviolet) flux and V (middle of the visible spectrum) flux. Similarly, V-J compares visible to infrared light. As you can see from the examples of stars above, comparing these three bands is a good way of finding the blackbody peak. So, you can use U-V vs. V-J to tell the difference between a galaxy with mostly massive (blue) stars and a galaxy with less massive (red) stars.
The other important thing about massive stars is that the rate at which they burn fuel goes up very rapidly: the luminosity of a main sequence star is proportional to M3.5, while the fuel supply only increases as M. Thus, the most massive stars might live only for around 100 million years or less, while a solar mass star has a lifetime closer to 10 billion.
When a galaxy is making a lot of stars, it makes them at all masses (searching for papers on the "initial mass function" is a good way to find details). So, we have both blue stars and red stars, and because the blue stars very luminous (remember, L ~ M3.5), the galaxy will look very blue.
Once a galaxy is no longer making stars, the massive ones die out, so only the lower-mass, redder stars remain. Thus, a quiescent galaxy will look red. And, as you might imagine, there are in-between galaxies, which have only stopped forming stars relatively recently, and are somewhere between the two.
So, using U-V vs. V-J is a good way of seeing whether the galaxy is mostly blue or mostly red, which tells us whether it formed stars recently.