I don't think it's great to use terms like "mix" or "hybrid" because people tend to think the screen is half E-Ink and half LCD, which isn't true.
Though the screens could indeed be called mixes or hybrids of LCD varieties, it's important to distinguish electrophoretic displays (EPD) from liquid crystal displays (LCD). They are completely different technologies.
For those who don't already know this, LCDs are not all backlit. Look at most digital watches, calculators, and many other devices and you'll see a variety of reflective LCDs. A higher-resolution reflective LCD panel can be found in the jetBook and Aluratek Libre readers. It relies on a mirror-like layer below the liquid crystal to bounce light through.
Backlit displays basically ditch the mirror layer and put lights behind the liquid crystals so you can see them in more diverse lighting conditions (particularly in the dark). However, the light needs quite a bit of power to transmit brightly through all the layers of an LCD. Since there's no reflective layer behind the liquid crystal layer, the screens will not display well when external lighting overpowers the backlight (like how you can't see out the window at night when all the indoor lights are on).
Pixel Qi is a mix or hybrid of transmissive (backlit) LCD and reflective (non-backlit) LCD. Screens somewhat like Pixel Qi have existed for years ("transflective" screens), but they are not as highly refined or redesigned as Pixel Qi is.
So when the backlight is on, you have a backlit display just like any other transmissive LCD. When you take it into bright light, the mirror layer present allows the screen to remain visible like reflective LCD, though you lose colors due to the position of the color filters in the design. The mirror is always there, whether the backlight is on or off, so in that sense it is a reflective LCD (which has always been a kind of e-paper)...but it gives you the option of toggling a backlight for color or viewability in lower light.
The advantages are that you get full LCD redraw speeds so you can handle video or other tasks that require rapid screen updates, you of course get the option of color, and you get a sort of resolution boost as the color subpixels in the normally 1024x600 color screen give you an effective monochrome resolution of 3072x600 for things like text, if the software supports subpixel rendering. With the backlight off, you reduce the screen's power consumption by around 80% as well.
Hope that helps.
Addendum: Oh, and they're now working on a 7" display in addition to their 10" model. Hopefully someone somewhere actually
uses it in one of their real