One less than DuoWB ;~).
The "uni" refers to the fact that this is a white balance with the red, green, and blue coefficients all set the same (to a value of 1, or at least something extremely near 1).
To understand what that means you have to understand that when a camera (or software) applies "white balance" to raw data, the green data is (generally) left as is and the blue and red values are swung to different values (the coefficient). In "warm" indoor light, for example, we have an abundance of red data, so we lower red and increase blue. In "cold" outdoor shade, we have an abundance of blue data, so we increase red and decrease blue.
Here's the thing: white balance is a multiplier (or divider) of the actual raw data. So you could have a raw data value of 250 (in 8-bit terms) for a red photosite in the sensor, but the final white balanced JPEG could push that value to 255 or down to 200. In the first case, it looks as if red is "blown" out, but the raw data was only 250. You might see that "blowout" on the histogram or highlight display and lower exposure in response, but you don't actually have a blowout.
Likewise, if the histogram seems to put values lower down (that 200 for our example red pixel), you might increase exposure thinking it is too low. But that 200 was an adjusted value in the first place—the real value was 250. Thus, if we increase exposure based upon the histogram, we risk blowing out that red pixel.
UniWB uses coefficients of 1 for red, green, and blue. If you multiply something by 1, it comes out to be the same value as it was before ;~). Thus, by using UniWB you get accurate histograms and highlights for your raw data.
So why don't camera makers allow a setting of UniWB? Don't get me started! Okay, I'm started... Nikon used to allow us to use Capture to create UniWB settings and move that over to the camera. With Capture NX they took that facility out. Hello? Anyone home there in Tokyo? Apparently not.
One reason why the camera makers are reluctant about UniWB is that the JPEG (and thus the embedded preview image that shows up on the LCD of the camera) will come out with a heavy green tint. Which just begs the issue of why they don't calculate histograms and highlights display from the raw data instead of the embedded JPEG. Short answer: it's easier and faster and the camera companies don't understand that photographers might actually want the correct data.
I told you not to get me started. Silly. Design. Decision. For people that buy my books on the pro and prosumer DSLRs, I used to supply a UniWB balance file that can be loaded and used until such time as the camera makers come to their senses (true for my older D90, D300/D300s, D700, D5100, D7000, D3/D3s/D3x books). Unfortunately, every camera needs a different file (and in some cases, every firmware revision, too), and the proper use of it needs a chapter of its own in my books, so it's a pain to have to keep creating these files and describing them.