The fourteen tithis
The Matsya Purana and other texts list them as: Ashwina Shukla Navami; Kartika Shukla Dwadashi; Chaitra Shukla Tritiya; Bhadrapada Shukla Tritiya; Phalguna Amavasya; Pausha Shukla Ekadashi; Ashadha Shukla Dashami; Magha Shukla Saptami; Shravana Krishna Ashtami; Ashadha Purnima; Kartika Purnima; Phalguna Purnima; Chaitra Purnima; and Jyeshtha Purnima.
Many of them are already festivals: Magha Shukla Saptami is Ratha Saptami, Shravana Krishna Ashtami is Janmashtami in many places, Ashadha Purnima is Guru Purnima, and Kartika Purnima is Dev Deepawali.
How they are kept
The texts say that a shraddha or tarpana offered to the ancestors on a Manvadi day satisfies them for a long time, and that bathing in a holy river and giving charity bring great merit. Many families who follow the full rites of the ancestors perform tarpana with sesame and water on these days, alongside the Amavasya and Sankranti offerings.
The fourteen Manus
The Manus are Svayambhuva, Svarochisha, Uttama, Tamasa, Raivata, Chakshusha, Vaivasvata (the present one), Savarni, Daksha Savarni, Brahma Savarni, Dharma Savarni, Rudra Savarni, Deva Savarni (Raucya) and Indra Savarni (Bhautya). Each Manvantara lasts about 306.72 million years, and has its own Indra, sages and avatars.
Manvadi and Yugadi
Together with the four Yugadi days, the fourteen Manvadi days make up the eighteen tithis that the texts single out for offerings to the ancestors outside Pitru Paksha. Some panchangs mark them in the calendar of each month.