Lunar day (Tithi), Paksha and how far it has elapsed.
The Tithi is the lunar day: the time for the Moon to gain 12° on the Sun, giving 30 tithis across a lunar month split into the Shukla (waxing) and Krishna (waning) paksha. It is one of the five limbs of the Panchāṅga and a key factor in Muhurtham selection.
A Tithi is a lunar day: the time the Moon takes to gain 12° of longitude on the Sun. There are 30 Tithis in a lunar month, and because the two bodies vary in speed a Tithi can run anywhere from about 19 to about 26 hours.
The lunar month splits into two halves of fifteen Tithis. Shukla Paksha is the waxing half, from new moon to full moon; Krishna Paksha is the waning half, from full moon back to new moon.
A Tithi begins and ends at the moment the 12° elongation is completed, which can be at any hour. The Tithi in force at sunrise is conventionally the one the whole day is named for.