The difference between 1 and 2 is that reconcile suggests middle-ground between 2 parties was reached - "They were able to reconcile their differences", while conciliate suggests that one party made a peaceful move towards the other "He made conciliatory advances".
3 and 4 describe a 3rd party in a dispute. 'Mediate' describes a middle-man assisting in negotiations, while 'arbitrate' describes a a position of judgement or authority:
"The US will mediate discussions between China and Russia."
"The teacher was called to arbitrate a disagreement between students."
Student or Learner