A Tithi (தினம், தேதி) is a lunar day in the Hindu Panchang. It is defined as the time it takes for the angular distance between the Sun and Moon to increase by 12°. Because the Moon moves at a variable speed in its elliptical orbit, a Tithi can last anywhere from about 19 to 26 hours — it does not correspond to a fixed clock duration.
There are 30 Tithis in a lunar month: 15 in the Shukla Paksha (waxing fortnight, from new moon to full moon) numbered Pratipada (1st) through Purnima (full moon), and 15 in the Krishna Paksha (waning fortnight, from full moon to new moon) numbered Pratipada (1st) through Amavasya (new moon).
Tithis are grouped into five categories based on their astrological quality: Nanda (1st, 6th, 11th — associated with joy, good for pleasure activities), Bhadra (2nd, 7th, 12th — good for steady work), Jaya (3rd, 8th, 13th — favours victory and competition), Rikta (4th, 9th, 14th — generally inauspicious), and Purna (5th, 10th, 15th — complete and auspicious for all activities).
Because a Tithi is based on angular motion and not clock time, a single Tithi can span two solar days (Tithi Vriddhi — the Tithi is "expanded") or a Tithi can be skipped entirely within a solar day (Tithi Kshaya — the Tithi is "reduced"). This is why the Panchang Tithi for a given day may be different from what you might expect if you count mechanically.
For auspicious ceremonies (Muhurtham), astrologers seek a favourable Tithi combined with an auspicious Nakshatra, good Vara, and outside inauspicious windows like Rahu Kalam. Purnima (full moon) is especially auspicious. Ekadashi (11th Tithi) is sacred to Lord Vishnu and observed as a fast day by Vaishnavas. Pradosham (13th Tithi in Krishna Paksha) is sacred to Lord Shiva.