The phase
The shape drawn in each day is the Moon as it stands at noon in your city, lit from the real angle of the Sun on that date.
How the phases are countedThe sign
The Moon crosses a sign of the zodiac in about two and a half days. The sign is the same everywhere on Earth at the same moment; only the local clock time changes.
The Moon in the signsThe lunar day
A count from one to thirty that starts at the new Moon and turns over at every moonrise, which is why it depends on where you are.
The thirty lunar daysPhases, signs, moonrises and moonsets are calculated in your browser with an open astronomy library and match published ephemerides to the minute. Each day shows the state of the Moon at 12:00 local time, and the lunar day is counted from moonrises in the city named at the top, so a city a few hundred kilometres away can differ by one. The names of the lunar days follow the Russian tradition set out by P. Globa; the sources page says what is calculated and what is tradition.