And the new PS4 controller is getting rid of the "analog face buttons" that were on the PS3 controller. What is that referring to? The Start/Select buttons?
In this context, "digital" buttons have two states - on and off. Meanwhile, "analog" buttons have multiple states, usually corresponding to how much pressure is being applied.
Playstation controllers from the PS2 and PS3 generations typically have pressure-sensitive face buttons. This means that the triangle, square, circle, and X buttons are not simply "on" or "off," but capable of detecting certain differences in the amount of force applied and reporting that to the game you're playing.
In a racing game, for instance, the throttle or brake controls might be mapped to one of these buttons, and the amount of pressure you apply determines how fast you speed up or slow down. If I recall correctly, GTA3 on the PS2 used this feature.
The original Xbox controllers also had this feature, but they dropped it in the Xbox 360, which breaks certain Xbox games that are otherwise compatible with the 360's software/hardware.
Most current-gen console controllers have at least a couple of other sets of analog inputs - the triggers, and the sticks. As the trigger has become more popular, the use of analog features of the face buttons has tended to decline.
For instance, in the PS2 era Dualshock 2, the R1/R2/L1/L2 were pressure sensitive buttons. In the PS3's Dualshock 3, the R2/L2 buttons became triggers instead.
This report is stating that these face buttons will likely be digital in the PS4's iteration of the controller.