Tripundra (Sanskrit: "three marks") or பட்டை / Pattai in Tamil is a saivite tilaka. 'Tripundra' refers to the three mukti. tri means three, pundra means one which is released. There are three horizontal lines of vibhuti (holy ash) on the brow, often with a dot (bindu) as the third eye. The three lines represent the soul's three bonds: anava, karma, and maya. Holy ash, made of burnt cow dung, is a reminder of the temporary nature of the physical body and the urgency to strive for spiritual attainment and closeness to God.
Actually pundarika is a name of Lord Vishnu which also means lotus. The lotus signifies purity which is devoid from any pollution. The "Tri" is what siginifies the triguna - Satva, Raja and Tama; Triloka - Bhuhu, Bhuvaha, Suvaha; Tritapa -Bbhoutika, Daivika, Adhyatmika.
It is the 'bhasma' dharana. Bhasma means one which has sacrificed everything which it has. For eg. that 'bhasma' the sacred ash has lost its original shape, size, color, fragrance, karma all and burned for the sake of some one and gave light, heat, food, rain everything by burning its own physical body. and it is called 'Vibhuti'. means that is the right meaning of GOD. He burns himself in the form of Sun to give light to the whole universe. The meaning goes like this elaborately.
One who wears 'tripundra' on his fore head should chant the mantras of lord at the same time mentally understand the meaning of that. Means what he is wearing on his forehead is not just a sign of sampradaya, culture, family explanations. But it has got more thing to imbibe culture in him, elevate his understandings and also learn to sacrifice and get things in his life.