Each day's moon phase is worked out from how many days have passed since a known new moon, divided by the length of one lunar cycle (about 29.53 days). That fraction tells us both how much of the Moon is lit and which of the eight named phases the day falls into.
A New Moon is fully dark; a Full Moon is fully lit. In between, the lit portion grows (waxing) from New Moon through First Quarter to Full Moon, then shrinks (waning) from Full Moon through Third Quarter back to New Moon. The calendar only labels the day each of these eight phases begins, to keep the grid readable.
Every day shows a small moon reflecting that day's actual illumination. A text label appears only on the day a new named phase begins. Times and the highlighted "today" cell use your device's local timezone.