The concept of Upanayanam is steeped in symbolism. A Brahmachari is allowed to wear a Yagnopaveeta of 3 strands, a householder 6 strands and a householder who has performed his Shastyaabdhapurti (60th birthday) is allowed to wear 9 strands. The 9 strands are also worn to signify the last 3 strands as a substitute for the Angavastram. Each 3 strands are tied together with a knot and only one knot is permitted per thread. As a rule, the upaveetha is to be hand woven by using cotton and drawing thin shreds and rolling them with the help of the sacred ash and consecrated water. When the desired thickness is achieved for each strand, they are knotted.
There is a Tantra aspect to the concept as well. The three strands represent the three main naDis viz., Ida, Pingala and Sushumna. The knot signifies the Bramha Granthi - the first knot encountered while raising the Kundalini shakti (the divine energy) situated in the Muladhara chakra. It is said that only after breaking past the Bramha Granthi, the student is free from animalistic urges and is set on the right path of salvation.
The second thread (of three strands) are worn for two main reasons:
Everyone looking at him will know that he is married (people in the early days used to cover their upper body only with a cloth - Angavastram).
The better half (wife) usually a homemaker, will not have the time required for Sadhana (practice) and therefore, the responsibility of the wife's emancipation is also on the husband's shoulder. Therefore, he wears the second upaveetha on behalf of the wife and grants her equal share of the merits derived from his practice and spiritual quest.
The third and final upaveetha is worn basically to tell the world that he is a Vriddha (senior citizen) and therefore will command respect and his blessings to be sought.
The thread is worn from left to right because the Ida nadi (signifying the moon) runs from the Muladhaara on the right and ends on the left side of upper part of the body (at the Ajna chakra and the thread can only rest on the shoulder and not any further up).
The Ida side is chosen because the Ida naDi controls the mental functions of the brain and is responsible for contemplation, meditation and worship.
It is to be noted that while performing the daily Sandhyavandanam, pranayama is an integral part and in almost all lineages, it is taught to start the Anuloma-Viloma pranayama (alternate nostril breathing) from the left nostril first as it draws in the prana in to the "cool" channel rather than the right nostril ("hot" channel).
During the Shraddha ceremony the thread is worn on the other side (from right shoulder dropping to the left hip). This is because the departed soul is invited to the ceremony to partake of the tarpana and pinda dana. Souls, as they do not posses a body have 360 degree vision and see everything (front, back, sideways, up and down) all at the same time. This causes objects to be mirrored and wearing the upaveetha in the reverse actually appears the correct way to the invited soul.
People of 3 of the 4 chaturashramas - Bramhacharya, Grihasta and Vanaprastha as a rule are supposed to wear the Yagnopaveetha. People who have taken up the Sanyasa ashrama (Turiya ashrama) are not allowed as they are supposed to be beyond the regular timetabled practices and are to be constantly engrossed in Samadhi and are not required to perform daily austerities (but many Sanyasis still perform their daily austerities to set an example to followers)
The Yagnopaveetha is to be worn by all the people of the first three varna ashramas - Brahmana, Kshatriya and Vaishya The fourth ashrama are not allowed as their prescribed profession is manual labour and hence would not have the time to follow prescribed rules and worship.
During the Upanayana ceremony, the student is initiated into the Gayathri mantra as the first mantra for his life long spiritual quest and final emancipation. The Gayathri mantra is a moksha pradana mantra and not meant for acquiring material benefits and hence this mantra is taught.