The quarter days are defined by the 4 main solar astronomical events. These are split between the 2 solstices which give us the longest and shortest days of the year and the 2 equinoxes which are the days where night and day are of equal length, 12 hours each on Earth. It is worth noting that these events are directly related to the astrological cycle. I know that this may be stating the obvious but all too often the obvious is overlooked.
I have to admit to having a soft spot for the equinoxes. Why ? I like the idea of a balance between light and dark but much more important is the fact that this is one of the few events which has the same meaning at the same time everywhere on the surface of planet earth. One brief moment of global equality.
On the diagram a seasonal name is given to only one of the four points, Yule. This is not due to any lack of festival names which take place at these times but simply to highlight the fact that there are no obvious Celtic names corresponding to these events. On the Wikipedia page the events are given the names Yule, Ostara, Midsummer, and Mabon. I find it a little curious that they didn't use Litha for Midsummer. Yule and Ostara seem to be Germanic in origin and I couldn't find much for Mabon or Litha. I'm pretty sure that the Celts were aware of these important events but for some reason we do not seem to have names for them.
The quarter days are the scientific face of the wheel of the year; they can be observed and measured.
The most important thing about the quarter days is that they are not linked to the civil calendar. Nothing personal you realise, it's just that the civil calendar is a bit of a shambles, or endearingly eccentric, depending upon your point of view.