Some prestigious families trace their descent to before the reforms of Surak.
As established in Spock's World (SW), the leader of a clan was historically - and still is, but it's unknown and likely varies to what extent - the oldest mother in the clan. These women hold the logical title 'Eldest Mother' or 'Oldest Mother.' T'Pau, for example, is the Eldest Mother of Spock's clan.
In modern times, (also est. in SW) there is also a role and title of 'Head of House,' which can be and is not always a role held by the Eldest Mother. The title of Head of House can be held by a man.
T'Pau, as an example, held both titles, but upon her death Sarek became Head of House and, as she chose to give her katra to Amanda, Amanda became the Eldest Mother.
It is unknown what the differences between clan, house and family are, or what the difference in duties and authority between Eldest Mother and Head of House is. Presumably, a clan tends to contain several families. Perhaps the Head of House is the leader of a family, and the Eldest Mother is the leader of an entire clan as a whole, with several Heads of House under her.
T'Pau, to Sarek and Amanda:
You do not need this gift, son of my house. You will be Head of House now, and you would have difficulty dealing with my katra, I think. No. I think we will do well together, my daughter. You have the necessary training from Seleya to manage the Gift once I have left it to you: and it will qualify you as Eldest Mother of the house, whatever others may say. Best to so manage matters.
It is unusual for an adult Vulcan to be unmarried, due to the biological imperative of the Time (pon farr) - it is a matter of life and death to have a mating bond.
Most families practice, for lack of a better English term, arranged betrothal of their children, but not all do.
(Est. The Vulcan Academy Murders (VAM)) The loss of a marriage bond from a spouse's death, especially a sudden death, has a devastating effect on Vulcans. If not given emergency psychic care by close relatives (sometimes with the support of a healer) a newly-widowed Vulcan will often themself die shortly afterwards from the shock.
(Est. VAM, SW) Remarriage is expected due to, again, biological imperative, and a choice not to remarry seems to be generally met with concern or grief, as it means the bereaved spouse will likely also die within the next few years.