The rites meant for the deities must be performed with devotion and those meant for the pitrs or fathers must be performed with faith. What is done with devotion is yajna and what is done with faith is sraddha. While performing the former, the sikha must be gathered into a knot and the sacred thread must rest on the left shoulder, and while performing the latter the sikha must be worn loose and the sacred thread must rest on the right shoulder.
The sikha and the sacred thread are meant for these two purposes. Sannyasins do not have either. When they renounce the world they also renounce the rites for the fathers and cease to worship a number of deities. They adore the Paramatman directly without any worldly desire in their hearts. The followers of other religions too wear neither a sikha nor a sacred thread and they worship the Supreme God directly [that is without going through the stages in which the various deities are worshipped].
Let me tell you about the two positions of the sacred thread while performing the rites for the celestials and the fathers. We must face the east as we conduct various rituals. The north is the direction in which we make the passage to the celestials. This path is called ""uttarayana". Our departed fathers reside in the south. The saint-poet Tiruvalluvar calls them "tenpulattar", those dwelling in the south. "Dakshinayana" is the way to the world of the fathers. Bhagavan Krsna speaks of the two paths in the Gita.
When we sit facing the east to perform rites for the pitrs, which shoulder is to the south? The right one. So the sacred thread must rest on it.
To do "pradakshina" means to go facing the south. (In majority of temples the raja-gopuram-the main entrance tower -is in the east. When you enter it and start circumbulating you will be facing the south. )
When we sit facing the east to perform rites for the gods our left shoulder is to the north. So the sacred thread must rest on it. When we are not engaged in either of these two rites- that is when we are doing our office work or something else- the sacred thread must not rest on either shoulder and must be worn like a garland. (No one seems to observe this rule in practice now. Except during the rites for the fathers, most people have their sacred thread resting on their left shoulder. )
For a general background, please see here