What is this type of digits-only LCD called?
It's a 7-segments display. Unlike the dot-matrix character displays kevlar refers to these are most often not intelligent module. Most dot matrix displays have an HD44780-compatible controller which you simply can write ASCII codes to, but a 7-segment LCD will often be just the glass, with connections for segments and a number of backplanes (often up to 4).

  

Driving LCDs can be awkward since they don't use just two levels, so you can't drive them with common digital logic.   



The best thing you can do is select a microcontroller with integrated LCD controller, which you can connect the display directly to, like the TI MSP430x4xx. Like most controllers this one also knows just segments; it isn't aware of digits or anything. (Great, first we had a dumb display, now we have a dumb driver as well!) There's reason for this. These LCD drivers are often used to drive custom LCDs which may be a mix of a numeric part, bar graphs and custom symbols. Such a symbol is also a single segment, so it makes no sense to talk about digits.  



This display has symbols like "battery" and "alarm clock", but also all text fields, like "AM", "PM" and "SNOOZE" are symbols consisting of a single segment (i.e. controlled by a single bit).  

Further reading
MSP430x4xx Family User's Guide. LCD controller is covered on p.709 ff.