The introduction of the tenth avatar of Lord Vishnu Kalki has to begin with an understanding of the yugas and the need for his arrival. In Hinduism, it is believed that there are four yugas namely Satya Yuga, Treta Yuga, Dwapara Yuga and Kali Yuga. These cycles operate just like any normal calendar but the measure of it is done in centuries. We are believed to be living in the Kali Yuga and have crossed 5000 years since the Kurukshetra war.
Kalki has to come to finish the evil in the world and the universe and to restore the balance in the cosmos. People believe that Kalki will come when the world has only evil left in it. He will slay the world of the bad, wrong and evil and start the cycle of the universe again.
Vyasa was a very smart man. He had written down a few things in the Vedas and it clearly mentions the things that would happen in Kali Yuga and how Kalki will come to finish all of it. He had mentioned that men and women would get married with just verbal agreement, one would cheat to grow and become successful in business, becoming rich would bring in respect, people would visit religious institutions just for the sake of doing it etc. By the end of the Kali Yuga, people will be three to four feet tall and would have a grey complexion. Their life expectancy would be reduced to twenty or even less.
Kalki, the slayer would then come on earth on his white horse Devadatta and with a Nandaka or Ratna Maru Sword and kill all the atheists. He will then leave as the next cycle of the yugas begins. He was born to Visnu Yasa and Sumati of Shambhala region. Kalki would marry the Padma who would be an incarnation of Lord Lakshmi.
The Kaliyug has been told to start right after the Mahabharata war. There are certain indications of it written in the old scriptures. By the second, third and fourth phase of the Kali Yuga, people will start to forget the name of the lord, stop offering yagnas and prayers to the lord, and start the gruesome act of cannibalism.
Kalki would lead the world towards dharma once again as he slays all those who are adharmi. He would lead earth to a golden age once again. He would be purifying the consciousness of all living beings. Prophecies have made it clear that the end of the world would happen with a pralay (flood).
Parshurama who is the sixth avatar of Lord Vishnu is said to become the teacher of the Kalki avatar which is the tenth avatar of the supreme lord. He would teach Kalki all about the Vedas so he can instil a sense of moral and understanding about good and evil in Kalki. He would also teach him warfare techniques and dhanurvidya so that he can fight evil properly.
After restoring peace and golden age back in the world, he will divide the world into kingdoms amongst his men and will leave for Vaikunth with his parents and begin his rule.