In Hindu mythology, ‘Vahana’ or vehicle on which God or Goddess mount is the representation of some extraordinary power. When Goddess Durga killed Demon king Mahisasur, she was offered a white lion that is symbol of strength, braveness and life.
It may also show that the lion is a symbol of uncontrolled animalistic tendencies (such as anger, arrogance, selfishness, greed, jealousy, desire to harm others etc.) and her sitting on it reminds us to control these qualities, so that we are not controlled by them.
The lion symbolizes strength, courage and leadership. In one word Dharma. The roar is considered as one of voices of Goddess. Maa durga indicates the success in life.
Mother Durga is the Goddess or the supreme energy (Shakti) and Lion or Tiger are like embodiments of power which are controlled by the Goddess.
The lion of Mother Durga represents “Dharma”. It is mentioned in several texts that the Dharma is “Swayam-eva-mrigendrataa” (Dharma is like a lion by itself). The lion represents power, will and determination. Mother Durga riding the lion symbolises her mastery over all these qualities. This suggests to the devotee that one has to possess all these qualities to get over the demon of ego.
Women are always regarded as soft and cool. They are never as angry as men. But to present women can also be as danger as a Lion, Mother Durga shown riding a Lion. This is the symbol of DARING and Shaktishali (powerful as men).