From Hindupedia, the Hindu Encyclopedia
Sanyasa ashrama is one of the four stages of man's life. However a sanyasi is technically outside the social fold. He is outside social divisions or categories, and its rules. He does not have any social bond as such, though he can raise his meal through accepting alms and by attending people's houses as a guest. A sanyasi is not supposed to stay long in any single place, he should be continuously moving so that he is not attached to any person or place. This is why a sanyasi is in fact called parivrajaka, one that roams and is never settled in a place. The single point goal of a sanyasi is liberation, and nothing else. He should have no desires, temptations, ambitions, other than liberation.
However there are places where a sanyasi can live, such as mathas (monasteries). A sanyasi should be moving and should not be living in the monastery throughout the year, and can be returning to it every year.
There are many traditions of monasteries in Hinduism. Broadly they agree upon renunciation, detachment concepts but are diverse in practices. The different schools have different traditions and different monasteries. For instance Sanyasis following Sri Vaishnava (like Sri Jeeyar) follow different practices than those following advaita-smarta like Kanchi Acaryas.
It is not necessary for a sanyasi to be affiliated to any of the monasteries. Ideally he could just be alone, a parivrajaka, with no fixed home.
Sanyasi and Society
Sanyasi has a peculiar kind of relation with the society. He can be dependent on it for his food, though not settled with one set of people. He can also teach them what he knows. However he shall not accept any assets from anyone for himself and can only do it to donate it or putting it to use for the benefit of the society (like doing yajna or giving it for building a monastery/temple/school etc). He can accept as disciples only sanyasis. He can however teach grhasthas (or brahmacaris or vanapratshis) though not take them with him.
Adi Sankara gave a small set of five poems called Yati pancaka, to explain how a Sanyasi should be. He gave two documents, for intra and inter-monastery (ashrama/math) activities. The first is to prescribe the way sanyasis should live in a monastery, their hierarchy and seniority, how they should treat each other, how the monastery should be organized etc. The second is about the way different monasteries should deal with each other. He laid down the dasanami sampradaya, where a sanyasi can be taking one of the ten suffixes to his name. They are bharati, sarasvati, sagara, tirtha, puri, asrama, giri, parvata, aranya and vana.
He established four mathas called Amnaya mathas in our sides of the country, at Puri (East), Sringeri (South), Dvaraka (West) and Jyotirmatha (North). They are associated with Rigveda, Yajurveda, Samaveda and Atharvaveda respectively. The dasanami sanyasins do not really have to belong to any particular matha and are arranged in to number of mathas apart from the Amnaya mathas. However the names are distributed within the Amnaya mathas as: vana and aranya with Puri, bharati, puri and sarasvati with Sringeri, tirtha and asrama with Dvaraka; sagara, parvata and giri with Jyotirmath.