Tarpana

Tarpana or is a term in the Vedic practice that refers to an offering made to divine entities. It refers to the act of offering as well as the substance used in the offering. Tilatarpana is a specific form of tarpana involving libations offered to the pitri using water and sesame seeds during Pitru Paksha or as a death rite.
Tarpana is a form of arghya. It is offered to all devas as well as the Navagrahas whenever mulamantra is recited as japa.

Instances of welcome ''tarpana''

For devis:

for Lakshmi, “आर्द्रां ज्वलंतीम् तृप्ताम् तर्पयन्तीम्” , "One who is satisfied and who satisfies those who offer Her Tarpana"
for Tripura Sundari, “बिंदु तर्पण संतुष्टाम् पूर्वजा त्रिपुराम्बिका” , "One who is satisfied by just a single drop of Tarpana"

Cow's milk, water, sugar, saffron, cardamom, borneo-camphor, etc. are mixed and used as the offering.
One tarpana is offered for the recitation of every ten mulamantras and one for part thereof.

''Tilatarpana''

Tilatarpana is the tarpana offered to pitrs by male descendants who do not have a living father. This is offered at the first annual shraddha and during subsequent annual shraddhas, amavasyas, sankramanas, eclipses, and during visits to selected holy places of pilgrimage.

Offering ''tilatarpana''

Black tila seeds, used along with water as the offering.
The yagnopavita is to be worn in the opposing position.
A pavitra made of kusha grass should be worn on the ring finger of the right hand and kusha grasses in left fingers horizontally called tekusha.
Posture of hand

The hand while offering tarpana varies for offerings made to devas, rishis and pitrus.

Gingelly

Gingelly is different than sesame, which is from a large tree; gingelly is a small seed that stores energy and hence is considered a favorite of all Devas, Shani and pitrus. Lord Ganesha is commonly offered pancha-kajjaya, a delicacy made using sesame. Lord Shiva is worshipped with sesame seeds. For Lord Vishnu, Lord Brahma, Goddess Laxmi and Goddess Saraswati, gingelly seeds are used in their favorite eatables. It is used as a homa dravya in many havans and homas. Therefore, tila and tilatarpana should not be dismissed as inauspicious.

Reasons for offering ''tarpana''

It is believed that one's pitrs eagerly await tarpana. If no offering is made, they return to their places disappointed, and the descendant misses their blessings that he would have received, if he had performed his filial duties. Kosha Kushi is an important ritual item used in the Tantric worship of the Divine Mother and represents the yoni and womb of the Goddess, as well as the astral body within the physical body and the microcosm within the macrocosm.