The Hindu texts are replete with references of deities using a particular animal as their primary vehicle. For example, Ganesha is associated with the rat, Shivji with an ox (Nandi), while Karthikeyan with a peacock, and Durga with a tiger. Is there an explanation behind the association of certain deities using a particular animal as the deity's vehicle? If so, what is the significance or the reasoning behind them?
As you pointed out Most of the Hindu Gods have their vehicles as animals.
I will explain about the Lord Ganesha and rat. One outsider who does not know about the inner meaning may simply laugh when saying that Lord Ganesha travel on a white rat.
Acutally Lord Ganesha is the God of Intelligence and attention. Before starting education(Vidyarambh) we pay tribute to Ganesha as 'Hari Sree Ganapathaye Nama'. His animal rat is the symbol of the human mind. It will never take rest, always mind is running(a running mind). So as a representation of Lord Ganesha(God of Intellect and attention)controlling the Mind, it has put it as like Rat is the vehicle of Lord Ganesha.
Similarly, every other Gods and Vehicle has it's own inner meaings. Like Shiva- Ox, Saraswathi- Hamsa, etc. Also apart from this there can be other explanations too, as mentioned in different stories in Puranas.
Although it is difficult to entire story of each vahana(vehicle) to god. I post some 2 of them which i know.
Ganesha - There was once a Gandharva (a celestial being) named Kroncha.In the court of Indra, he once met the sage Vamadeva, and filled with pride, insulted him. This incited the anger of the sage, who cursed him that he would lose the good looks he was so proud of, and become a mouse. Kroncha found himself turned into a mouse, and ashamed of his appearance, ran away. He entered the ashram of Sage Parashara, and, true to his nature in his new form, started creating havoc. Meanwhile, Ganesha was paying a visit to the sage. The sage’s disciples approached Ganesha and asked him to put an end to the mouse which was wreaking havoc in their well kept ashram.Ganesha at once realized the true identity of the mouse. Using a rope, he lassoed the mouse, and mounted it. Contact with the divine body of Ganesha made Kroncha at last realize the error of his ways. Besides, Ganesha was so heavy that he felt himself being crushed. He prayed to Ganesha, “Lord, please forgive me for my sins. I have learnt my lesson. Please accept me as your mount and release me from this misery!” Ganesha relented and made himself light enough for the mouse to bear his weight. And thus did he become Mooshika Vahana.
Muruga A.K.A Karthikeya - Surapadma and his kin asuras terrorize Swarga, the place of Indra.Years of deva-asura battle had finally enraged Surapadma to the point of unleashing his full might in battle, a fist of fury that sought to catch hold of the devas and crush them once and for all. He had them imprisoned in the deepest and darkest dungeons of his city in Veeramahendrapuram. Simhamukha, the younger brother of the asura king Surapadma, came running into the throne room of his brother's palace, agitated and out of breath.As per the order of Lord Shiva, Karthikeya gets the "Vel" (A spear like weapon) and raged war against the both Surapadma and Simhamukha. This fierce battle ended with both of them being defeated.(In Hinduism, this battle is called as SooraSamharam). After this both of them needs salvation for their sins.Simhamukha and Surapadma were both blessed by Karthikeya to remain immortal. While Simhamukha was to become the lion steed of the goddess Kaali, Surapadma chose to serve Karthikeya himself. The king of the asuras offered himself to Murugan in two forms: a flagstaff bearing what was to be known thence as Karthikeya's sigil- a rooster; and Karthikeya's own mount, the peacock.
@Awal Garg is correct about Lord Shiva.
Durga - after being completely defeated by Mahisasura, the gods, headed by Brahma,approached Visnu and prayed for the destruction of the demon. Thereupon all the gods including Brahma, Visnu and Siva emitted flames of anger which solidified into the effulgent form of a goddess (Durga) with three eyes and ten arms,each of which bore a weapon given by the mighty gods. During this time the Himavat offered a white lion to goddess Durga as her vahana.
They all have stories behind them. Gods don't just randomly chose animals from a list. I will cite few examples1 below:
- Ganesha - While Ganesh was still a child, a giant mouse began to terrorize all his friends. Ganesh trapped him with his lasso and made him his mount. Mushika was originally a gandharva, or celestial musician. After absentmindedly walking over the feet of a rishi (wise man) named Vamadeva, Mushika was cursed and transformed into a mouse. However, after the rishi recovered his temper, he promised Mushika that one day, the gods themselves would bow down before him. This came to pass when Mushika's path crossed Ganesh's.
- Shiva - Before becoming the vehicle of Shiva, Nandi was a deity called Nandikeshvara, lord of joy and master of music and dance. Then, without warning, his name and his functions were transferred to the aspect of Shiva known as the deity Nataraja. From half-man, half-bull, he became simply a bull. Since that time, he has watched over each of Shiva's temples, always looking towards him.
1. Source : Wikipedia - Vahana.