Origins
A council is, at its etymological core, a summoning. The word descends from Latin concilium — 'an assembly, a meeting' — composed of con- ('together') and calāre ('to call, to summon'). To hold a council was to call people together for deliberation.
Latin calāre is the same root behind calendar. Roman priests called out (calāre) the first day of each month — the calends — and from this practice the calendar took its name. The connection between councils and calendars is that both begin with a public announcement: someone calls, and others respond.
The most persistent confusion in English is between council and counsel. They sound alike, they appear in similar contexts, and they were frequently interchanged in medieval manuscripts. But they are different words. Council (the assembly) comes from concilium. Counsel (advice) comes from cōnsilium, a different Latin compound meaning 'deliberation'. A council gives counsel, which does nothing to help the spelling.
Latin Roots
Reconcile belongs to this family too. Latin reconciliāre meant to bring together again — to restore a broken council, to reassemble what had fractured. The 'concil' in reconcile is the same concilium.
Church councils — Nicaea, Chalcedon, Trent — used the word in its most formal Latin sense: a body called together under authority to settle doctrine. English local councils preserve the same structure, if not the same grandeur.