Most of the deities in Hinduism are seen riding animals as their vehicles. It is believed that the positive qualities of these animals represent the qualities that the riding deity looks for. Such as the bull of Shiva represents strength and virility, the mouse of Ganesha represents the speed and sharpness.
Similarly grace, wisdom and beauty are represented by swan, the vehicle of Goddess Saraswati, and splendor &majesty by the peacock, the vehicle of Lord Skanda. This depiction of Gods with animals is found in the Greek mythology as well.
Surya Dev is the Lord of Sun. He is worshiped for earning prosperity and fame. Surya Dev rides on a chariot driven by seven white horses. These horses represent the seven sins, which are overpowered by Surya Dev. They are sometimes also associated with the seven chakras in the human body because Surya Dev blesses his devotees with good health as well.
Lord Brahma rides on a swan. A swan is known for its ability to pick pure from the impure. It can sieve milk from water. Due to this quality, it is often associated with intelligence. Lord Brahma therefore, is believed to ride on a swan.
The Goddess Durga rides a lion. A lion symbolizes strength and power. Just as the lion is a warrior, and always the victorious one. Goddess Durga, through it, also symbolizes immense power and victory. In many images the Goddess with a lion is shown killing a demon with a bull. She fights with all the demonic negativities, being assisted by this lion.
Ganesha rides on a mouse. According to one story, a Gandharva was once cursed to become a mouse. Being a mouse he started wandering on earth and by mistake he stepped on the foot of Mahari Vamadeva. A sage is considered a spiritual being in Hinduism. Enraged by this Lord Ganesha attacked him with his Pasha Astra.
This signifies that stepping on knowledge, awareness and enlightenment is considered sinful. Some stories associate a mouse with darkness and desires. Therefore Ganesha overrides all these leading the man to awareness and spiritual upliftment.
Lord Indra rides the Airavat elephant. It is a holy elephant which is believed to be born from a cosmic egg. Of the total sixteen elephants from that egg, this was the strongest. Hindu scriptures say, that this elephant drinks water from the atmosphere with its trunks and sprinkles that water on the earth. This water is seen as rain on the earth.
The elephant is the father of other white elephants. However, once it was cursed and lost its wings. It is believed that all the white elephants on the earth are his descendants.
Goddess Lakshmi mounts the white owl. Owl is associated with wisdom, and peace. It is also bestowed with the qualities of future telling. The white colour of the owl is related to purity of the soul. Hence, the Goddess rules the powers of wisdom, intelligence, purity and wealth.
Goddess Saraswati is the consort of Lord Brahma. She is seen with a swan. Moreover, the swan with its white colour and intelligence, can easily be associated with Goddess Saraswati.
It is believed that the bull Nandi had lived with Lord Shiva on mount Kailash. This animal is mainly associated with strength. Lord Shiva rides the bull. Nandi is seen seated besides Lord Shiva in many of the depictions. He is also worshiped along with the deity.
Lord of death, Yamaraja, is associated with righteousness and hence is his mount, the male buffalo. Lord Yamaraja roams around the world on a buffalo searching for souls he has to take with him.